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I 


AN  ELEMENTARY  TREATISE 


ON 


DYNAMICS. 


FIFTH    EDITION. 

AN  ELEMENTARY  TREATISE 

ON 

THE    INTEGRAL    CALCULUS, 

CONTAINING 

APPLICATIONS  TO  PLANE  CURVES  AND  SURFACES. 

BY 

BENJAMIN  WILLIAMSON,  F.  E.  S. 


SIXTH    EDITION. 

AN  ELEMENTARY  TREATISE 

ON 

THE    DIFFERENTIAL    CALCULUS, 

CONTAINING 

THE  THEORY  OE  PLANE  CURVES. 

BY 

BENJAMIN  WILLIAMSON,   F.R.S. 


AN     ELEMENTARY    TREATISE 


DYNAMICS, 


CONTAINING 


APPLICATIONS  TO  THERMODYNAMICS, 


NUMEROUS     EXAMPLES 


BY 

BENJAMIN  WILLIAMSON,  M.A.,  F.R.S., 

FELLOW  OF  TRINITY   COLLEGE,  AND  PROFESSOR   OF  NATURAL  PHILOSOPHY 

IN   THE   UNIVERSITY  OF    DUBLIN; 

ANT) 

FRANCIS  A.  TARLETON,  LL.D., 

FELLOW    AND    TUTOR     OF     TRINITY    COLLEGE. 


SECOND  EDITION,  REVISED  AND  ENLARGED. 


NEW   YORK: 
D.    APPLETON    AND    COMPANY.. 

1889. 


DBPT . 

DUBLIN  : 

PRINTED    AT    THE    UNIVERSITY    PRESS, 

BY  PONSONBY  AND  WELDRICK. 


<&4  ZrVr 

W  7 

!  k 

PHYSICS  DEPf, 


PREFACE. 


Although  in  recent  years  several  important  works  on 
Dynamics  have  been  published  in  England,  yet  none  have 
been  issued  which  seem  to  fill  the  role  contemplated  in  this 
book.  In  its  composition  we  have  started  from  the  most  ele- 
mentary conceptions,  so  that  any  Student  who  is  acquainted 
with  the  conditions  of  Equilibrium  and  with  the  notation  of 
the  Calculus  can  commence  the  Treatise  without  requiring 
the  previous  study  of  any  other  work  on  the  subject.  The 
first  half  contains  a  tolerably  full  treatment  of  what  is 
usually  styled  the  Dynamics  of  a  Particle.  The  latter  half 
treats  of  the  Kinematics  and  Kinetics  of  Eigid  Bodies ;  and 
throughout  we  have  kept  the  practical  nature  of  the  subject 
in  view,  and  have,  in  general,  avoided  purely  fancy  problems. 
In  an  early  chapter  we  have  introduced  and  elucidated 
the  general  principle  of  "Work  or  Energy,  and  have  given 
subsequently  a  more  complete  treatment  of  this  great 
principle,  illustrating  it  by  a  brief  application  to  the 
theory  of  Thermodynamics.  In  the  latter  part  of  the  book 
we  have  borrowed  largely  from  Thomson  and  Tait's  Natural 
Philosophy;   Routh's  Rigid  Dynamics;    Schell's  Theorie  der 


665465 


vi  Preface. 

Bewecjung  und  der  Krtifte ;  and  Clausius'  Mechanical  Theory  of 
Heat;  our  aim  having  been  simply  to  enable  the  Student  to 
acquire  as  easily  as  possible  a  knowledge  of  the  subject  of 
which  we  treat. 

In  this  Edition  we  have  carefully  revised  and  to  a  con- 
siderable extent  rearranged  the  entire  Work.  In  doing  so  we 
have  developed,  and  in  some  cases  rewritten,  many  por- 
tions of  the  subject,  more  especially  that  on  generalized 
coordinates  in  connexion  with  Lagrange's  and  Hamilton's 
methods.  We  have  also  exhibited  the  general  theory  of 
small  oscillations  in  a  new  form,  and  one  which  we  hope 
will  be  easily  comprehended  by  the  Student. 

To  those  who  desire  to  pursue  the  study  of  Dynamics  to 
its  highest  development,  the  perusal  of  the  great  treatise  of 
Thomson  and  Tait,  as  also  that  of  Routh,  will,  we  hope,  be 
facilitated  by  using  the  present  Work  as  an  introduction. 

We  may  add  that  to  the  latter  writer  our  obligations, 
as  the  reader  will  find,  have  been  largely  increased  in  this 
Edition. 


Teinity   College, 
i%,  1889. 


TABLE   OF   CONTENTS 


CHAPTEE    I. 


VELOCITY. 

PAGE 

Uniform  Motion, 

.       2 

Variable  Motion, 

.       3 

Kinematics, 

.       5 

Composition  of  Velocities, 

7 

Relative  Velocity,    . 

.       9 

Examples, 

.     11 

CHAPTEE    II. 

ACCELERATION. 

Uniform  Acceleration,      .         .         . .12 

Variable  Acceleration, 1» 

Accelerations  Parallel  to  Fixed  Axes,        .         .         .         ■         •         ■         .16 

Total  Acceleration,  .         .         .         . 17 

Tangential  and  Normal  Accelerations,       .         .         .         .         •         •         .17 

Hodograph, 19 

Angular  Acceleration,       .         .         .         .         -         •         •         •         •         .19 

Areal  Acceleration, -1 

Accelerations  Parallel  to  Moving  Axes,     .  22 

Units  of  Time  and  Space, 23 


viii  Table  of  Contents. 


CHAPTER    III. 

LAWS   OF   MOTION. 
Section  I. —  'Rectilinear  Motion. 

PAGE 

First  Law  of  Motion, 25 

Second  Law  of  Motion, 26 

Mass, 32 

Motion  on  a  Smooth  Plane, 34 

Line  of  Quickest  Descent  to  a  Curve,        .         .  ■ 36 

Section  II. — Parabolic  Motion. 

Construction  of  Path, 39 

Eange  and  Time  of  Flight, 40 

Morin's  Apparatus, 46 

Section  III. — Friction. 

Elementary  Laws  of  Friction, 50 

Motion  on  a  Rough  Inclined  Plane, 51 

Section  IV. — Momentum. 

Force  measured  by  Momentum, 53 

Absolute  and  Gravitation  Units, 54 

Impulses, 56 

Equations  of  Motion, 56 

Section  V. — Action  and  Reaction. 

Third  Law  of  Motion, 58 

Forces  of  Inertia, 59 

Atwood's  Machine, 60 

Examples, 64 


CHAPTER  IY. 

IMPACT    AND    COLLISION. 

Direct  Collision  of  Homogeneous  Spheres,         ......  66 

Height  of  Rebound, 69 

Oblique  Collision, 70 

Vis  Viva  of  a  System, 7;* 


Table  of  Contents.  ix 


PAGE 


Momentum  of  a  System, 74 

Conservation  of  Momentum,   .........       75 

Examples, .78 


CHAPTER  V. 

CIRCULAR    MOTION. 
Section  I. — Harmonic  Motion. 


Uniform  Circular  Motion, 84 

Elliptic  Harmonic  Motion, 86 

Section  II. — Centrifugal  Force. 

Circular  Orbits, 90 

Centrifugal  Force  of  Earth, .91 

Verification  of  the  Law  of  Attraction, 92 

Centrifugal  Force  in  Eotation  of  a  Rigid  Body, 94 

Section  III. — Motion  in  a  Vertical  Circle. 

Motion  in  a  Vertical  Curve, 98 

Simple  Pendulum,  .         .         .         .         .         .         .         •         •         .100 

Time  of  a  Small  Oscillation, 101 

Seconds'  Pendulum,       ..........     102 

Effect  of  Change  of  Place, 104 

Airy  on  Mean  Density  of  Earth, 107 

Time  of  Oscillation  for  any  Amplitude,   .         .         .         -         -         -         .108 
Cycloidal  Pendulum,       .         .         .  .         .  .  .  -         -  .111 

Conical  Pendulum,         .         .         .         .         .         .         .         •         •         .115 

Eevolution  in  a  Vertical  Circle, •         -     117 

Examples,      .         .         .         .         .         •         .         •         •         •         •         .122 


CHAPTER  VI. 


WORK    AND    ENERGY. 

Gravitation  Unit  of  Work, 125 

Absolute  Unit  of  Work, I'26 

Work  by  a  Variable  Force, 128 

Potential  of  a  Sphere, .         .  130 


X 


Table  of  Contents. 


PAGE 

Work  by  a  Stress, 131 

Energy, 133 

Kinetic  Energy, 133 

Equation  of  Energy, 136 

Energy  of  Eotation, 137 

Work  Done  by  an  Impulse, 140 

Compound  Pendulum,  . .141 

Motion  Round  a  Fixed  Axis,  .         . 144 

Examples,      . 145 


CHAPTER    VII. 

CENTRAL   FORCES. 
Section  I. — Rectilinear  Motion. 


Centre  of  Force, 147 

Force  varying  as  the  Distance, 148 

Force  varying  as  Inverse  Square  of  Distance, 149 

Application  to  the  Earth, 151 

Application  to  Spheres,  .         . 152 

Application  to  Elastic  Strings, 155 

Secion  II. — Central  Orbits. 

Differential  Equations  of  Motion, 160 

Law  of  Direct  Distance,  .         .         . 161 

Equable  Description  of  Areas, 164 

^        .      dhi  F 

Equation  — —  +  u  =  rrr—z, 171 

u  dd-  h~  u2 

Law  of  Inverse  Square, 173 

Kepler's  Laws, 175 

Law  of  Gravitation, 176 

Velocity  at  any  Point  in  Orbit, 177 

Change  of  Absolute  Force, 179 

Application  of  Hodograph, 181 

Lambert's  Theorem,       ..........  183 

Mass  of  Sun, 186 


Table  of  Contents.  xi 

PAGE 

Mean  Density  of  Sun,    .         .         .         -         .         .         .         •         •         .187 
Planetary  Perturbations,  .         .         .         .         •         •         •         •         -     188 

Tangential  Disturbing  Force, 189 

Normal  Disturbing  Force,       .         .         -         •         •         •         •         •         .189 

Apsides, 190 

Approximately  Circular  Orbits, 194 

Movable  Orbits,  Newton, 196 

Examples,      .....-..•-••     197 


CHAPTER  VIII. 

CONSTRAINED    MOTION,    RESISTING    MEDIUM. 

Motion  on  a  Fixed  Curve, 206 

Theorem  of  Ossian  Bonnet,     .....••••  -08 
Motion  on  a  Fixed  Surface,    .         .         .         .         •         •         •         •         .211 

Motion  on  a  Sphere,       .......•••  212 

Rectilinear  Motion  in  a  Resisting  Medium, 219 

Examples, 223 


CHAPTER  IX. 

THE    GENERAL   DYNAMICAL    PRINCIPLES. 
D'Alembert's  Principle, 227 


Initial  Motion, 


230 


Bertrand's  Theorem, 23i 

Thomson's  Theorem, 23! 

Equation  of  Vis  Viva, 232 

Effect  of  Impulses  on  Vis  Viva, 23'^ 

Equations  of  Motion, 24° 

Constraints  and  Partial  Freedom, 24i 

Moments  of  Momentum,         .         .         .         .         •         •         •         •         -243 

Conservation  of  Moment  of  Momentum,           ...-••  2^8 

Examples, 248 


xii  Table  of  Contents. 

CHAPTER  X. 

MOTION    OF    A    RIGID    BODY   PARALLEL   TO    A    FIXED    PLANE. 
Section  I. — Kinematics. 

PAGE 

Degrees  of  Freedom, 254 

Translation  and  Eotation, 255 

Composition  of  Finite  Displacements,      .......  256 

Composition  of  Velocities,       .........  257 

Space  Centrode  and  Body  Centrode, 261 

Pure  Rolling, 261 

Geometrical  Representation  of  Motion,    .......  262 

Examples, 263 

Section  II. — Kinetics — Coyistrained  Motion. 

Degrees  of  Freedom, 268 

Motion  Round  a  Fixed  Axis, 270 

Moments  of  Momentum,  .         .         .         .         .         .         .         .         .273 

Stresses  on  Axis  of  Rotation, 274 

Stress  Due  to  Impulses, 275 

Centre  of  Percussion, 276 

Examples, 279 

Section  III. — Kinetics  of  Free  Motion  Parallel  to  a  Fixed  Plane. 

Equations  of  Motion, 283 

Equation  of  Vis  Viva, 285 

Moment  of  Momentum,  relative  to  any  Point,          .....  286 

Impact, 288 

Stress  in  Initial  Motion, 292 

Friction, 296 

Tendency  of  a  Rod  to  Break, 302 

Impulsive  Friction, 308 

Roiling  and  Twisting  Friction, 311 

Examples 312 


Table  of  Contents.  xiii 


CHAPTER  XI. 


MOTION    OF    A    RIGID    BODY    IN    GENERAL. 
Section  I. — Kinematics. 

PA.GE 

Composition  of  Rotations,        .         .         .         .         .         .         .         .         .317 

Motion  of  a  Body  entirely  Free,      ........  320 

Analytical  Treatment  of  Motion, 321 

Velocity  of  any  Point  of  a  Body,    ........  325 

Acceleration  of  Rotation, 327 

Complete  Determination  of  Motion  of  a  Body, 329 

Screws, 333 

Composition  of  Twists, 334 

Examples, 335 


Section  II. — Kinetics. 

Moments  of  Momentum  round  a  Fixed  Point,  .         .         .         .         .315 

Motion  of  a  Body  Round  a  Fixed  Point  under  the  Action  of  Impulse.      .     346 
Couple  of  Principal  Moments,         .         .         .         .         .         .         .         .347 

Motion  of  a  Free  Body  under  Impulse,   .......     350 

Vis  Viva  of  a  Rigid  Body,      .         .         .         .         .         .         .         .         .351 

Equations  of  Motion  of  a  Body  having  a  Fixed  Point,     ....     353 

Equations  of  Motion  of  a  Free  Body, 354 

Motion  of  a  Body  Round  a  Fixed  Point  under  no  External  Force,    .         .     357 
Conjugate  Ellipsoid  and  Conjugate  Line,  ......     363 

Stress  exerted  by  a  Body  on  a  Fixed  Point,      ......     367 

Centrifugal  Couple, 369 

Motion  relative  to  Centre  of  Inertia,        .         .         .         .         .         .         .371 

Impact,  ............     379 

Impulsive  Friction, 380 

Collision  of  Rough  Spheres,  381 

Equations  of  Motion  Referred  to  Body-axes, 385 

Motion  consisting  of  Successive  Rotations,       ......     385 

Motion  of  a  Solid  of  Revolution, 389 

Examples, 390 


xiv  Table  of  Contents. 


CHAPTER  XII. 

ENERGY    AND    THE    GENEEAL    EQUATIONS    OF    DYNAMICS. 
Section  I. — Energy. 

PAGE 

Equation  of  Energy, 396 

Conservation  of  Energy, 397 

On  the  Ultimate  Permanent  Forces  of  Nature, 399 

Forces  which  appear  in  the  Equation  of  Energy, 400 

General  Form  of  Equation  of  Energy, 402 

Equivalent  Systems  of  Forces,        ........  404 

Wrenches,     ............  405 

Examples,      ............  405 

Section  II. — The  General  Equations  of  Dynamics. 

General  Equations  of  Motion, 412 

Equation  of  Energy  when  Conditions  Involve  the  Time,  .         .         .413 

Similar  Mechanical  Systems, 414 

Generalized  Coordinates, .         .         .415 

Kinetic  Energy  in  Generalized  Coordinates, 416 

Generalized  Equations  of  Motion  under  Impulse,     .         .         .         .         .416 
Generalized  Expression  for  Kinetic  Energy,    .         .         .         .         .         .419 

Energy  of  Initial  Motion, 420 

Lagrange's  Generalized  Equations  of  Motion,  .....     420 

Deduction  of  Equation  of  Energy, 423 

Ignoration  of  Coordinates, 426 

Components  of  Momentum  and  Velocity,         ......     429 

Hamilton's  Form  of  Equations  of  Motion, 431 

Calculus  of  Variations, 433 

Examples, 433 

Principle  of  Least  Action, 436 

Hamilton's  Characteristic  Function, 439 

Examples 442 


Table  of  Contents.  xv 


CHAPTEE   XIII. 


SMALL     OSCILLATIONS. 

PAGE 

Oscillation  on  a  Plane  Curve, 445 

Oscillation  on  a  Surface,         ......                  .         .  446 

Conditions  for  Stable  Equilibrium, 451 

Equations  of  Motion  for  an  Oscillating  System, 452 

General  Solution  of  these  Equations, 454 

Harmonic  Determinant,  .         .         .         .         .         .         .         .         .454 

Lemma  in  Determinants, 455 

Transformation  of  Harmonic  Determinant, 457 

Reality  of  Roots  of  Harmonic  Determinant,    ......  457 

Stability  of  the  Motion, .         .'        .459 

Case  of  Equal  Roots, 462 

General  Solution  in  this  Case,         . 464 

Principal  Coordinates  and  Directions  of  Harmonic  Vibration,           .         .  465 

Effect  of  an  Increase  of  Inertia, 469 

Energy  of  an  Oscillating  System, 470 

Examples,      ............  471 


CHAPTER    XIY. 


THERMODYNAMICS. 


Mechanical  Equivalent  of  Heat,      .         .         .         .         .  .         .477 

Equation  of  Energy,       ..........  478 

Specific  Heat, 479 

Perfect  Gas, 481 

Reversibility  and  Cyclical  Processes, 483 

Isothermals  and  Adiabatics  for  a  Perfect  Gas,           .....  484 

Fundamental  Principles  of  Thermodynamics,           .....  485 

Carnot's  Cycle, 486 


xvi  Table  of  Contents. 

PAGE 

Extension  of  Camot's  Cycle, 488 

Entropy, 489 

Energy  and  Entropy, 490 

Elasticity  and  Expansion, 491 

Examples, 492 

Non-reversible  Transformations,     . 495 

Examples, 496 

Absolute  Scale  of  Temperature, 497 

Absolute  Zero, 499 

Change  of  State, 501 

Examples, 504 

Available  Energy, 507 

Dissipation  of  Energy, 508 

Increase  of  Entropy, 509 

Path  of  Least  Heat, 511 

Examples, 512 

Miscellaneous  Examples 514 


DYNAMICS. 


CHAPTER    I. 


VELOCITY. 


1.  Matter. — "We  give  the  name  of  matter  to  that  which 
exclusively  occupies  space,  and  which  we  regard  as  the 
permanent  cause  of  any  of  our  sensations.  Portions  of 
matter  which  are  bounded  in  every  direction  are  called  bodies. 
Every  body  has  necessarily  a  determinate  volume,  and  an 
external  form  or  surface ;  and  exists,  or  is  conceived  to  exist, 
in  space. 

A  portion  of  matter  indefinitely  small  in  all  its  dimen- 
sions is  called  a  material  particle.  Every  body  may  be  re- 
garded as  consisting  of  an  indefinitely  great  number  of 
particles.  The  name  of  force  is  given  to  any  cause  which 
produces,  or  tends  to  produce,  motion  in  matter.  The  branch 
of  Mechanics  which  treats  of  motion  produced  in  a  body  by 
the  action  of  force  is  commonly  called  Dynamics. 

"We  commence  with  the  consideration  of  motion  in  itself, 
without  any  regard  to  its  cause. 

2.  Motion,  Velocity. — When  a  body  continually 
changes  its  position  in  space,  it  is  said  to  be  in  motion ; 
and  the  rate  and  the  direction  of  the  motion  of  any  of  its 
points  at  any  instant  is  called  the  velocity  of  the  point  at 
that  instant. 

The  motion  of  a  point  is  said  to  be  rectilinear  or  curvilinear 
according  as  its  path  is  a  right  line  or  curved. 

In  the  case  of  curvilinear  motion,  the  direction  of  motion 
of  a  particle  at  any  instant  is  that  of  the  tangent  to  its  path, 
drawn  at  the  point  occupied  by  the  moving  particle  at  the 
instant. 


2  • '  ''   'Velocity. 

3.  Motion  0f  TFr&n^lation. — If  all  the  points  of  a  rigid 
body  move,  at  each  "instant,  in  parallel  directions,  the  body  is 
said  to  have  a  motion  of  translation  only ;  and  the  motion  of 
the  body  is  completely  determined  when  that  of  any  one  of  its 
points  is  known.  It  is  usual,  in  this  case,  to  take  its  centre 
of  mass  as  the  point  whose  motion  determines  that  of  the 

hody.  .  . 

In  our  earlier  chapters,  whenever  we  speak  of  a  rigid 
body  moving,  we  suppose  it  to  have  a  motion  of  transla- 
tion solely,  and  we  consider  its  path  as  that  of  its  centre  of 
mass. 

4.  Uniform  Motion,  Velocity. — If  a  point  move  over 
equal  lengths  or  spaces,*  in  equal  intervals  of  time,  however 
short  the  intervals  be  taken,  its  motion  is  said  to  be  uniform  ; 
and  its  velocity  is  measured  by  the  space  described  in  the  un  it 
of  time :  this  is  the  same  at  every  instant  so  long  as  the 
motion  continues  uniform. 

A  second  is  usually  adopted  as  the  unit  of  time  ;  and,  in 
this  country,  a  foot  as  the  unit  of  length.  Thus,  the  velocity 
of  a  point  which  moves  over  five  feet  in  each  second  is  said  to 
be  a  velocity  of  5  feet  per  second,  and  is  numerically  denoted 
by  5  ;  and  similarly  in  other  cases.  If  any  other  units  of 
time  and  space  be  adopted,  the  number  which  represents  the 
velocity  of  the  moving  point  will  have  to  be  altered  pro- 
portionally. Thus,  we  speak  of  a  velocity  of  10  miles  an 
hour,  or  100  yards  a  minute,  &c. :  each  of  these  can  be  readily 
expressed  in  feet  per  second,  when  necessary. 

The  space,  or  length  of  the  path  described  during  any 
time,  is  usually  denoted  by  the  letter  s,  the  velocity  by  v,  and 
the  time  estimated  in  seconds  by  t.f  In  the  case  of  uniform 
motion,  the  relation  connecting  these  quantities  can  be  imme- 
diately obtained.  For,  if  the  space  described  in  one  second 
be  represented  by  v3  that  described  in  two  seconds  is  repre- 
sented by  2v,  that  in  three  seconds  by  3r,  and  that  in  any 
number  (t)  of  seconds  by  vt. 


*  The  -word  space  is  employed  in  abbreviation  for  length  of  path  described. 

f  Unless  the  contrary  be  stated,  we  shall  in  all  cases  assume  a  foot  and  a 
second  as  our  units  of  space  and  time,  i.e.  we  shall  regard  t  as  representing 
a  number  of  seconds  or  parts  of  a  second,  and  s  as  a  number  of  feet. 


Variable  Motion.  3 

Accordingly  we  have  in  the  case  of  uniform  motion  the 
relation 

s  =  vt.  (1) 

This  formula  evidently  holds  good  whatever  be  the  units 
of  space  and  time,  and  introduces  the  unit  of  velocity  as  that 
of  a  unit  of  space  described  in  a  unit  of  time.  It  is  true  for 
uniform  curvilinear,  as  well  as  rectilinear  motion  ;  and  also 
whether  t  represents  a  number  of  seconds,  or  any  part  of  a 
second,  however  small. 

Again,  if  s  denote  the  space  described  in  the  time  t\  we 
have    «'  =  vtf,     and  hence 

s'-s 

or  the  velocity,  when  uniform,  is  measured  by  the  space  de- 
scribed during  any  interval  of  time  divided  by  the  number  by 
which  that  time  is  represented. 

This  result  equally  holds  good  if  we  suppose  the  interval  of 
time,  denoted  by  if  -  t,  to  become  indefinitely  small ;  in  which 

case  the  limiting  value  of  -, — -  or  —  will  still  represent  the 

v    —  L  CIO 

velocity  v. 

Examples. 

1.  If  a  body,  moving  uniformly,  pass  over  10  miles  in  an  hour,  find  its  ve- 
locity in  feet  per  second.  Ans.  14|, 

2.  If  a  body,  moving  uniformly  with  a  velocity  of  16  feet  per  second,  pass         V 
over  100  miles,  find  the  time  of  its  motion.  Ans.  9  hrs.  10  min. 

3.  Assuming  that  light  travels  from  the  sun  to  the  earth  in  8m  30s,  and  that         ^ 
its  velocity  is  180,000  miles  per  second,  calculate  the  distance  of  the  sun. 

Ans.  91,800,000  miles. 

4.  If  a  velocity  of  20  miles  an  hour  be  the  unit  of  velocity,  and  a  mile  the        v" 
unit  of  space,  find  the  number  which  represents  a  velocity  of  32  feet  per  second. 

Ans.  \r£' 
b.  Find  in  metres  the  velocity  of  a  point  on  the  earth's  equator  arising  from 
the  rotation  of  the  earth  on  its  axis.  Ans.  463. 

5.  Variable  Motion. — If  the  spaces  described  in  equal 
intervals  of  time  be  not  equal,  the  motion  is  said  to  be 
■variable,  and  the  velocity  can  no  longer  be  measured  by  the 
space  actually  described  in  one  second.  The  movable  has, 
however,  at  each  instant  a  certain  definite  velocity  which  is 

b2 


4  Velocity. 

measured  by  the  space  which  it  ivould  describe  during  a  second, 
if  it  were  conceived  to  move  uniformly  during  that  time  with  the 
velocity  which  it  has  at  the  instant  under  consideration. 

For  example,  when  we  say  that  a  railway  train  is  moving 
at  the  rate  of  40  miles  an  hour,  we  mean  that  it  would  pass 
over  40  miles  in  the  hour  if  it  continued  to  move  during  that 
time  with  the  speed  which  it  has  at  the  instant  referred  to. 

Again,  if  we  suppose  that  there  are  no  sudden  changes  of 
velocity,  the  change  in  the  velocity  of  a  movable  in  any  in- 
definitely small  portion  of  time  must  be  itself  indefinitely 
small;  as  otherwise  the  velocity  would  not  vary  continuously. 
Accordingly,  in  such  cases,  we  may  suppose  the  motion  as 
uniform  during  the  indefinitely  small  time  dt;  and  we  shall 
have  (as  in  the  last  Article)  for  the  velocity  v  at  any  instant 
the  equation 

s  -  s      ds 

v  m  lm-  7=i  =  If  (2) 

That  is,  in  all  cases  the  velocity  of  a  point  at  any  instant  is 
measured  by  the  limiting  value  of  the  space  described  in  a 
small  interval  of  time,  divided  by  the  number  which  repre- 
sents that  interval  of  time.  This  method  of  expressing  velo- 
city is  sometimes  concisely  represented  in  the  notation  of 
Newton  by  the  symbol  s. 

6.  Mean  Telocity. — If  a  body  describe  the  space  s  in 
the  time  t,  then  its  mean  or  average  velocity  during  that  time 

is  represented  by  -,  being  the  velocity  with  which  a  body, 

V 

moving  uniformly,  would  describe  the   same   space  in  the 

time  t.     The  formula  (2)  can  be  immediately  deduced  from 

the  consideration  of  mean  or  average  velocity — for  we  may 

consider  the  velocity  of  a  point  at  any  instant  as  being  its 

mean  velocity  during  an  infinitely  small  interval  of  time; 

ds 
whence  we  get,  as  before,  the  relation  v  =  — . 

OjTi 

7.  Geometrical   Representation  of  a  Velocity. — 

Uniform  rectilineal  motion  is  completely  determined  when 
the  direction  and  rate  of  motion  are  known.  Hence  the 
velocity  of  a  point  can  be  represented  both  in  magnitude  and 
direction  by  a  right  line. 


Kinematics.  5 

Thus,  if  a  point  move  uniformly  in  the  line  OP,  so  as  to 

describe  the  space  OA  in  the  unit 

of  time  (one  second  suppose),  the 
line  OA  may  be  taken  to  repre- 
sent the  velocity  of  the  point  both  in  magnitude  and  direc- 
tion.     The  arrow  head  denotes  the  direction  in  which   the 
motion  takes  place,  namely  from   0  to  A. 

This  method  of  representation  holds  good  also  in  the  case 
of  variable  velocity,  provided  OA  be  the  space  which  the  body 
would  describe  in  one  second  if  its  velocity  remained  unaltered 
in  magnitude  and  direction  (Art.  5). 

In  accordance  with  the  principles  established  in  Geometry, 
if  the  velocity  of  a  particle  moving  from  0  to  P  be  regarded 
as  positive,  velocity  in  the  opposite  direction,  i.  e.  from  P  to  0, 
must  be  regarded  as  negative. 

8.  Kinematics. — As  our  ideas  of  motion  and  velocity 
depend  solely  on  our  conceptions  of  space  and  time,  the  whole 
subject  of  motion  admits  of  being  treated  as  a  branch  of  pure 
Mathematics ;  and,  as  such,  has  been  discussed  in  many 
important  treatises  during  recent  years. 

This  branch  of  Mathematics  is  called  Kinematics*  (from 
Kivr/jua,  motion),  and  in  it  the  motion  of  a  body  is  discussed 
without  any  reference  to  the  force  or  forces  by  which  the 
motion  is  produced.  Questions  of  the  latter  class,  i.  e.  of 
motion  with  reference  to  force,  belong  to  the  science  of  Dy- 
namics, or  what  is  now  usually  styled  Kinetics. 

The  foregoing  distinction  should  be  observed  by  the 
student,  as  much  indistinctness  of  conception  arises  from  its 
not  being  carefully  kept  in  mind  in  the  study  of  Dynamics. 

In  the  present  treatise  it  is  not  proposed  however  to  divide 
the  treatment  of  the  subject  in  the  manner  indicated,  as  to 
do  so  would  require  a  complete  discussion  of  motion  (in- 
cluding rotation  and  kindred  subjects)  before  entering  on  the 
most  elementary  problems  in  Dynamics.  At  the  same  time 
it  will  aid  the  student  towards  obtaining  clear  mechanical 
conceptions  if  he  will  consider   what  part  of  each  problem 


*  The  name  "  Cinematique  "  was  first  given  to  this  hranch  of  Mathematics 
by  Ampere,  in  his  "  Essai  sur  la  philosophie  des  Sciences,"  1834 


6  Velocity. 

discussed  belongs  properly  to  the  science  of  Kinematics,  and 
what  to  that  of  Dynamics  or  Kinetics. 

9.  Rest  and  Motion,  Relative. — We  have  defined 
rest  and  motion  with  reference  to  space.  Now  of  space  in 
itself  or  absolute  space  our  senses  take  no  cognizance,  all  that 
we  perceive  being  matter  or  body  as  occupying  or  existing  in 
space ;  but  our  senses  give  us  no  information  as  to  whether 
any  body  occupies  the  same  absolute  position  in  space  during 
successive  intervals  of  time  or  not.  Hence,  of  absolute  rest 
we  can  have  no  perception  or  knowledge ;  and  when  we  say 
that  a  body  is  at  rest  we  mean  that  it  does  not  alter  its  posi- 
tion with  relation  to  other  bodies  which  are  considered  fixed. 
For  instance,  bodies  on  the  earth's  surface  are  said  to  be  at 
rest  when  they  do  not  alter  their  position  relatively  to  the 
earth's  surface  ;  we  know  however  that  the  earth  has  at  least 
two  distinct  motions,  one  of  rotation  relative  to  its  axis ;  the 
other  around  the  sun,  regarded  as  fixed.  As  our  idea  of  rest 
is  only  relative,  so  also  must  be  our  idea  of  motion  :  thus,  a 
body  is  said  to  be  in  motion  when  it  alters  its  position  with 
respect  to  other  bodies  regarded  as  being  at  rest. 

Hence  all  motions  must  be  considered  as  relative :  for  in- 
stance, when  we  say  that  a  body  is  moving  at  the  rate  of 
thirty  miles  an  hour,  we  mean  that  such  is  its  velocity  relative 
to  a  place  on  the  earth :  its  absolute  velocity  is  immensely 
greater,  and  is  obtained  by  combining  this  velocity  with  the 
absolute  velocity  of  the  earth  itself. 

Again,  we  speak  of  the  same  body  as  at  rest,  or  as  in 
motion,  according  as  we  compare  its  position  with  that  of  one 
object  or  of  another.  For  example,  a  person  seated  in  a 
railway  carriage  is  said  to  be  at  rest  relatively  to  the  carriage, 
and  to  be  in  motion  relatively  to  the  earth,  &c. 

That  a  body  may  be  regarded  as  having  at  the  same  in- 
stant two  or  more  velocities  is  a  matter  of  common  experience : 
for  instance,  if  a  ball  roll  along  the  deck  of  a  vessel,  which  is 
descending  a  river,  we  conceive  the  ball  as  having  simul- 
taneously one  velocity  along  the  deck ;  another,  that  of  the 
vessel  in  the  stream  ;  a  third,  that  of  the  river  relatively  to 
its  banks,  &c.  The  velocity  of  the  ball,  relatively  to  the 
earth,  is  got  by  compounding  these  separate  velocities.  We 
proceed  to  show  in  what  manner  this  can  be  done. 


Composition  of  Velocities.  7 

10.  Composition  of  Velocities. — Suppose  a  point  to 
move  uniformly,  with  a  velocity  v, 
along  the  line  AB,  while  the  line 
moves  uniformly  parallel  to  itself ; 
then  the  point  may  be  regarded  as 
having  the  two  velocities  simulta- 
neously. In  order  to  find  its  position 
at  the  end  of  any  time  t,  let  AB  be 
the  space  which  it  would  describe  in  that  time  along  AB 
considered  as  fixed;  and  let  CD  be  the  position  of  the 
moving  line  at  the  end  of  the  same  time;  complete  the 
parallelogram  ABDC;  then  D  will  plainly  be  the  position 
of  the  moving  point  at  the  end  of  the  time  t.  Also,  if  v  be 
the  uniform  velocity  of  the  point  along  the  line  A  C,  we  shall 
have  AC  =  v't,  and  CD  =  vt.     Hence 

AC  jf 

CI)~  v' 

Again,  as  this  is  independent  of  t,  the  ratio  of  AC  to  CD 
will  be  constant  during  the  entire  motion  ;  and  consequently 
the  point  will  move  from  A  to  D  along  the  diagonal  AD. 

To  find  the  velocity  of  the  moving  point,  we  make  t  =  1 
(or  the  unit  of  time)  in  the  last;  then  AB  and  AC  represent 
in  magnitude  and  direction  the  component  velocities^  of  the 
moving  point,  and  AD  represents  the  resultant  velocity  :  in 
other  words,  if  a  body  be  animated  by  two  velocities  repre- 
sented in  magnitude  and  direction  by  the  sides  of  a  parallel- 
ogram, the  resultant  velocity  is  represented  in  magnitude  and 
direction  by  the  diagonal  of  the  parallelogram. 

Conversely,  any  velocity  may  be  regarded  as  equivalent 
to  two  velocities  in  any  two  directions,  and  the  magnitudes  of 
the  component  velocities  can  be  determined  by  the  preceding 
construction. 

In  like  manner,  if  a  body  be  animated  simultaneously 
with  three  velocities,  its  resultant  velocity  is  represented  in 
magnitude  and  direction  by  the  diagonal  of  the  parallelepiped 
whose  edges  represent  the  component  velocities.  For  we  can 
compound  two  of  these  velocities  by  the  method  given  above, 
and  then  compound  their  resultant  with  the  third  velocity. 
This  principle  can,  plainly,  be  extended  to  the  case  of  a  point 


8  Velocity. 

supposed  to  be  animated  by  any  number  of  velocities  simul- 
taneously. 

11.  Polygon  of  Velocities. — It  immediately  follows 
that  if  a  point  be  subjected  to  any  number  of  simultaneous 
velocities  its  resultant  velocity  can  be  obtained  by  the  fol- 
lowing geometrical  construction  : — 

From  0,  the  original  position  of  the  point,  draw  OA, 
representing  one  of  the  given  velocities  in  magnitude  and 
direction  ;  from  A  draw  AB,  parallel  and  equal  to  the 
line  which  represents  a  second  velocity ;  and  so  on  for  the 
remaining  velocities ;  then  the  line  which  connects  0  with 
the  extremity  of  the  line  drawn  parallel  and  equal  to  the 
line  representing  the  last  velocity  will  represent  the  resultant 
velocity,  both  in  magnitude  and  direction. 

This  construction  is  called  the  polygon  of  velocity,  and  is 
in  general  a  gauche  polygon. 

The  preceding  result  admits  of  being  stated  otherwise, 
thus  :  If  a  body  be  subjected  to  two  or  more  uniform  veloci- 
ties it  will  arrive  at  the  same  position  at  the  end  of  any  time 
as  it  would  have  arrived  at  if  the  several  motions  had  taken 
place  successively  instead  of  simultaneously.  This  is  adopted 
as  an  axiom  by  some  writers  on  Mechanics,  for  it  appears  to 
be  an  immediate  consequence  of  our  ideas  of  motion.  The 
student  can  easily  see  that  the  whole  theory  of  the  composi- 
tion of  velocities  can  be  deduced  from  this  principle. 

12.  Component  and  Resultant  Velocities. — The 
velocities  represented  by  AB  and  AC,  in  Art.  10,  are  called 
the  components  of  the  velocity  represented  by  AD. 

If  a  point  describe  a  plane  path,  the  usual  method  of 
representing  its  position  is  with  reference  to  two  fixed  rect- 
angular axes  lying  in  the  plane. 

Then,  if  cc,  y  be  the  coordinates  of  the  moving  point  at 
any  instant,  its  component  velocities  parallel,  respectively,  to 
the  coordinate  axes,  are  evidently,  by  Art.  5,  represented  by 

dx      ,  clu 
■ —  and  —  • 
dt         dt 

Also,  if  a  be  the  angle  which  the  direction  of  motion  at 
the  instant  makes  with  the  axis  of  x,  the  component  veloci- 
ties are  represented  by  v  cos  a  and  v  sin  a,  respectively;^',  e. 
the  velocity  with  which  a  point  is  moving  in  any  fixed  direc- 


Relative  Velocity.  9 

tion  is  equal  to  the  component  of  its  velocity  in  that  direc- 
tion. 

TT  i  dx  .  du 

Hence  we  get    v  cos  a  =  — ,     Psma  =  -J.  (3) 

If  we  square  and  add,  we  get 

dt)  +\it)  ~\Jt);  •'•  v~Jt; 

i.e.  the  velocity  in  a  curvilinear  path  is  represented  in  the 
same  matter  as  in  a  rectilinear  ;  this  result  might  have  been 
directly  established  from  oflier  considerations. 

More  generally,  if  a?,  i/,  z  be  the  coordinates  of  a  moving 
point  at  any  instant,  with  reference  to  any  system  of 
coordinate  axes,    its    component  velocities    parallel  to   the 

coordinate  axes  are  plainly  represented  by  — ,  -~  and  — ,  re- 

at   at  etc 

spectively.     If  the  axes  be  rectangular,  and  if  o,  |3,  7  be  the 

direction  angles,  and  v  the  magnitude  of  the  velocity  of  the 

point,  then  the  component  velocities  parallel  to  the  coordinate 

axes  are  represented  by  v  cos  a,  v  cos  |3,  v  cos  7,  respectively. 

Hence,  in  this  case,  we  have 

dx  _     du  dz  ... 

v  cos  a  =  — ,    v  cos  p  =  -£,    v  cos  7  =  — .  (4) 

dt  '       dt  '      dt  x 

In  Newton's  notation,  as  in  Art.  5,  these  component 
velocities  are  represented  by  the  symbols,  xt  i/,  z. 

13.  Relative  Velocity. — If  the  point  A  be  in  motion 
along  AB  with  a  velocity  represented 
by  AB,  and,  at  the  same  time,  A!  be 
in  motion  along  A'B'  with  a  velocity  re- 
presented by  A'B\  to  find  their  relative 
velocity. 

Draw  AD  parallel  and  equal  to  A'B\ 
and  construct  the  parallelogram  ACBD; 
then  the  velocity  AB  may  be  regarded 
as  equivalent  to  the  velocities  AD  and 
AC;  now  the  former  velocity,  being  equal  and  in  the  same 
direction  as  that  of  the  other  point  A',  will  not  alter  the  relative 


10  Velocity. 

position  of  the  points  (Art.  10)  ;  consequently  the  latter  com- 
ponent AC  represents  the  relative  velocity  of  the  moving 
points,  i.e.  the  velocity  with  which  A  is  moving  relatively  to 
A'9  regarded  as  at  rest. 

Hence,  to  get  the  velocity  of  one  moving  point  relatively 
to  another  which  is  also  in  motion,  we  suppose  equal  and 
parallel  motions  given  to  both,  each  equal  and  opposite  to  the 
motion  of  the  second  point:  by  this  means  that  point  is  brought 
to  rest,  and  the  velocity  of  the  other,  relative  to  it,  is  had  by 
compounding  the  new  velocity  with  its  original  velocity. 

14.  Components  of  Relative  Telocity. — Suppose 
(x,  y,  z),  (/,  y',  z)  to  be  the  coordinates  of  the  two  moving 
points  (M,  M'),  respectively,  with  reference  to  any  coordi- 
nate system  of  fixed  axes.  Then,  to  get  the  motion  of  M\ 
relatively  to  31,  we  suppose  three  axes  drawn  through  M 
parallel,  respectively,  to  the  coordinate  axes ;  and  let  £,  rj,  £ 
denote  the  coordinates  of  M,  relative  to  these  axes,  and  we 
have 

£  =  %'  -  x,  ri  =  y'-y,  Z  =  s'  -  s ; 
and  hence 

d%      dxf      dx      dr)  _  dy       dy      a%  _  dz'      dz  ^        _ 
di  =  dt~It'    It  =  dt  "  di'    di'pdi'di'     ^' 


i.e. 


dx'     dx     df      dy      dz       dz 
dt  ~  di'     di  "df    Jt  " 'di' 


or  x  -  x,     y  -  y, 


represent  the  components  of  the  relative  velocity  of  the  two 
moving  particles. 


Examples.  1 1 


Examples. 

1.  Two  points  are  moving  in  rectangular  directions,  with  velocities  of  300 
and  400  yards  per  minute  ;  find  their  relative  velocity  in  feet  per  second. 

Ans.  25. 

2.  Two  particles  start  simultaneously  from  different  points,  in  givendirec- 
tions,  with  uniform  velocities.  Show  how,  hy  a  geometrical  construction,  to 
determine  the  relative  distance  at  the  end  of  any  time ;  and  find  when  this 
distance  is  a  minimum. 

3.  The  tide  is  running  out  of  the  mouth  of  a  harbour  at  the  rate  of  2^  miles 
per  hour ;  in  what  direction  must  a  man,  who  can  row  in  still  water  at  the  rate 
of  5  miles  per  hour,  point  the  head  of  the  boat  in  order  to  make  for  a  point 
directly  across  the  harbour  ? 

4.  A  boat  starts  with  a  given  velocity  across  a  river  ;  find  the  direction  in 
which  she  should  steer,  in  order,  without  altering  her  course,  to  land  at  a  given 
station  at  the  opposite  side  of  the  river — the  velocity  of  the  stream,  and  also  of 
the  boat,  being  supposed  known. 

5.  Two  trains  are  moving,  one  due  south,  the  other  north-east. ^  If  their 
velocities  be  25  and  30  miles  an  hour,  respectively,  calculate  their  relative 
velocity. 

6.  A  railway  train  is  moving  at  the  rate  of  30  miles  an  hour,  when  it  is 
struck  by  a  stone,  moving  horizontally  and  at  right  angles  to  the  train  with  the 
velocity  of  33  feet  per  second.  Find  the  magnitude  and  direction  of  the  velo- 
city with  which  the  stone  appears  to  meet  the  train. 

Ans.  Resultant  velocity  is  55  feet. 
Indian  Civil  Service  Exam.,  1876. 

7.  Two  particles  start  simultaneously  from  A,  JB,  two  of  the  angular  points 
of  a  square  ABC  I),  in  the  directions  AB,  BC;  and  describe  the  periphery  with 
constant  velocities  V,  v,  respectively,  where  V  is  greater  than  v,  until  one  par-      y 
ticle  overtakes  the  other.     Prove  that  the  minimum  distances  between  the  par- 
ticles occur  at  equal  intervals  of  time,  and  that  if  V  :  v  :  :  m  +  1  :  m,  where  m 

is  an  integer,  the  sum  of  all  these  minimum  distances  is 

m  (m  +  1)  . ,      .  ,, 

x  a  side  of  the  square. 


2y/wH(»»+  l)a 

Camb.  Math.  Trip.,  1871. 


12  Acceleration. 


CHAPTEE    II. 

ACCELERATION. 

15.  Acceleration  and    Retardation    of  Motion. — The 

velocity  of  a  point  is  said  to  be  accelerated  or  retarded 
according  as  it  increases  or  diminishes  with  the  time.  This 
acceleration,  or  rate  of  change  of  velocity  in  a  fixed  direction, 
may  be  either  uniform  or  variable.  Retardation  of  motion 
is  to  be  regarded  as  a  negative  acceleration,  i.e.  as  an  accelera- 
tion in  the  opposite  direction  to  that  of  the  motion. 

16.  Uniform  Acceleration. — The  motion  of  a  point 
moving  in  a  straight  line  is  said  to  be  uniformly  accelerated 
when  it  receives  equal  increments  of  velocity  in  equal  times. 
In  this  case  the  acceleration  is  measured  by  the  additional 
velocity  received  in  each  unit  of  time.  As  a  second  is  usually 
taken  as  the  unit  of  time,  we  may  define  the  acceleration  of 
velocity  in  this  case  to  be  measured  by  the  additional  velocity 
received  by  the  movable  in  each  second;  this  acceleration  is 
usually  denoted  by  the  letter  /. 

In  the  case  of  uniform  acceleration  in  a  right  line  we 
proceed  to  find  expressions  for  the  velocity  at  the  end  of  any 
given  time,  and  also  for  the  space  described. 

17.  Velocity  at  any  Instant. — Let  v0  denote  the  velo- 
city at  the  instant  from  which  the  time  is  reckoned;  then, 
since  the  point  receives  in  each  second  an  additional  velocity 
/,  its  velocity  at  the  end  of  the  first  second  is  v0  +  /;  at  the 
'end  of  the  next  second,  v0  +  2/;  at  the  end  of  the  third,  v0  +  3/; 

and  at  the  end  of  n  seconds,  v0  +  nf.  Or,  if  ^  t  denote  the 
number  of  seconds  in  question,  and  v  the  velocity  at  the  end 
of  that  time,  we  have 

V  =  Vo  +ft.  (1) 

If  the  point  be  supposed  to  start  from  rest,  we  have 
v=ft; 


Space  described  in  any  Time.  13 

that  is,  the  velocity  acquired  at  the  end  of  t  seconds  is  t  times 
that  acquired  at  the  end  of  one  second. 

In  the  case  of  a  "uniformly  retarded  motion,/ denotes  the 
Telocity  lost  in  each  second ;  and,  if  v0  he  the  initial  velocity, 
we  shall  have,  as  before,  for  the  velocity  at  the  end  of  t 
seconds, 

v  =  v0-ft.  (2) 

In  this  case  the  velocity  becomes  zero  at  the  instant  when 
r0  =ft,  or  at  the  end  of  the  time  -^  •  If  the  retardation  con- 
tinued afterwards,  the  velocity  would  become  negative ;  that 
is,  the  point  should  proceed  to  move  back  in  a  direction 
opposite  to  that  of  its  former  motion. 

It  will  be  observed  that  the  formula  (1)  and  (2)  differ 
only  in  the  sign  of/;  they  may  accordingly  be  regarded  as 
comprised  in  the  same  general  formula,  in  which  a  retarda- 
tion, as  stated  before,  is  regarded  as  a  negative  acceleration. 

Examples. 

1 .  If  a  body  start  from  rest  with  a  uniform  acceleration  of  7  feet  per  second, 
find  its  velocity  at  the  end  of  three  minutes. 

Ans.  1260  feet. 

2.  In  what  time  would  a  body  acquire  a  velocity  of  100  feet  per  second  if 
it  start  from  rest  with  a  uniform  acceleration  of  32  feet  per  second  ? 

Ans.  Z\  seconds. 

3.  A  body  starts  from  rest  with  the  velocity  of  1000  feet  per  second,  and  its 
motion  is  uniformly  retarded  by  a  velocity  of  16  feet  each  second  ;  find  when  it 
would  be  brought  to  rest. 

Ans.  1  m.  2 -J  sec. 

4.  A  velocity  of  one  foot  per  second  is  changed  uniformly  in  one  minute  to 
a  velocity  of  one  mile  per  hour.  Express  numerically  the  rate  of  change  of 
velocity  when  a  yard  and  a  minute  are  taken  as  the  units  of  space  and  time. 

Ans.  -3*. 

18.  Space  described  in  any  Time. — To  find  the 
space  described  in  any  time  in  the  case  of  uniform  accelera- 
tion in  a  straight  line. 

From  equation  (2)  we  get 
ds 

hence,  by  integration, 

s  =  v0t+  I  ft2 ; 


\J 


14  Acceleration. 

no  constant  being  added  since  the  space  is  measured  from  the 
position  of  the  point  when  t  =  0. 

If  the  point  start  from  rest  we  have 

s  =  ift\ 

In  the  case  of  uniformly  retarded  motion  we  have 

s  =  v0t  -  iff-. 

This  and  the  preceding  formula  are  represented  by  the 
single  expression 

s  =  v0t±ift\  (3) 

in  which  the  upper  or  lower  sign  is  given  to  /,  according  as 
the  acceleration  has  place  in  the  positive  or  negative  direc- 
tion. 

Similarly,  equations  (1)  and  (2)  are  combined  in  the  state- 
ment 

v  =  v0±ft.  (4) 

The  preceding  result  admits  also  of  being  established  geo- 
metrically in  the  following  manner,  as  given  by  Newton  : — 

Suppose  the  point  to  start  from  rest,  and  on  any  right 
line  AX  take  portions  AD,  AE,  &c, 
proportional  to  the  intervals  of  time 
from  the  commencement  of  the 
motion,  and  erect  perpendiculars 
DB,  EC,  &c,  representing  the 
corresponding  velocities ;  then  since 
the  velocity  at  the  end  of  any  time  (Art.  18)  is  proportional 
to  that  time,  the  ordinates  BD,  CE,  &c,  will  be  to  one  another 
in  the  same  ratio  as  the  times,  i.  e.  as  ADy  AE,  &c. ;  and 
consequently  the  points  Af  B,  C,  &c,  all  lie  on  a  right 
line. 

Again,  let  AD  =  t,  DE  =  At,  BD  =  v;  then  the  space 
described  in  the  infinitely  small  time  At  will  be  represented 
by  vAt,  i.  e.  by  the  area  BDEC;  and  accordingly  the  whole 
space  described  in  the  time  represented  by  AN  will  be  repre- 
sented by  the  sum  of  the  elementary  areas,  BDEC,  &c,  or 
by  the  whole  area,  APN,  i.  e.  by  I  AN x  PN,  or  by  \vt ; 
therefore  s  =  \ft",  as  before. 

If  the  point  be  supposed  to  start  with  an  initial  velocity 


Variable  Acceleration.  15 

r0,  the  student  will  find  no  difficulty  in  supplying  the  corre- 
sponding construction. 

19.  Relation  between  Velocity  and  Space. — If  we 

eliminate  t  between  equations  (3)  and  (4),  we  get 

v-  =  iv  ±  2/s,  (5) 

in  which  the  upper  or  lower  sign  is  taken  according  as  the 
acceleration  is  in  the  direction  of  the  motion  or  in  the  oppo- 
site direction. 

We  shall  resume  the  consideration  of  these  equations 
when  we  come  to  the  investigation  of  the  motion  of  a  body 
under  the  action  of  a  constant  force. 

20.  Algebraic  Expression  for  an  Acceleration. — 
In  the  case  of  a  point  moving  with  a  uniform  acceleration, 
let  v  represent  the  velocity  at  the  end  of  the  time  t,  and  v 
that  at  the  time  tf;  then  by  (1)  we  have 

v  =  v0+ft,     v'=v0+ft', 

and  hence  /  =  j, — -. 

Moreover,  since  this  result  holds,  however  small  the  in- 
terval of  time  represented  by  f  -  t  may  be,  we  have,  as  in 
Art  4, 

civ 
J  ~  dt' 

21.  Variable  Acceleration. — In  the  case  of  the  motion 
of  a  point  in  a  right  line,  if  the  acceleration  is  not  uniform,  but 
varies  continuously  according  to  any  law,  we  plainly  (as  in 
Art.  5)  may  suppose  that  the  motion  is  uniformly  accelerated 
during  an  infinitely  small  time  dt ;  or  (which  is  the  same 
thing)  that  the  acceleration  at  any  instant  is  measured  by 
what  the  increase  of  velocity  in  a  unit  of  time  would  hare  been 
if  its  rate  of  increase  had  been  uniform  during  that  time,  and  the 
same  as  that  at  the  instant  in  question.  Hence  the  accelera- 
tion at  any  instant  is  defined  as  the  rate  of  change  of  the  velo- 
city at  that  instant,  and  is  measured  in  all  cases  by  the  ratio 
of  the  increment  of  the  velocity  at  the  instant  to  the  incre- 
ment of  the  time. 


16  Acceleration. 

Accordingly  we  have,  whether  the  acceleration  be  uniform 
or  variable,  the  relations 

/=*?«*?.  (6) 

J      dt      df-  K  j 

These  are  expressed  in  Newton's  notation  in  the  form 

/  =  b  =  s. 

All  these  results  apply  equally  to  the  case  of  retardation 
of  motion,  which  is  always  to  be  regarded  as  a  negative  acce- 
leration. 

22.  Geometrical  Representation  of  an  Accelera- 
tion.— From  the  preceding  it  appears  that  the  acceleration 
of  the  motion  of  a  point,  whether  it  be  uniform  or  variable, 
is  in  all  cases  measured  by  a  velocity.  Hence  it  can  be  re- 
presented, both  in  magnitude  and  direction,  by  a  right  line,  in 
the  same  manner  as  velocity  (Art.  7). 

Hence,  also,  we  may  regard  a  point  as  receiving  two  or 
more  simultaneous  accelerations  of  motion,  and  can  deter- 
mine the  resultant  acceleration  by  a  geometrical  construction, 
as  in  Arts.  10  and  11. 

Consequently,  accelerations  are  compounded  and  resolved 
according  to  the  same  laws  as  velocities. 

23.  Component  Accelerations  Parallel  to  Fixed 
Axes. — If  x,  y,  z  denote  the  coordinates  relative  to  a  fixed 
rectangular  system  of  axes,  of  the  position  of  a  moving 
point  at  the  end  of  the  time  t ;  then,  as  in  Art.  12,  its  com- 
ponent velocities  parallel  to  the  axes  of  coordinates  are  re- 

.    _  .      dx      dy      dz  ..     1 

presented  by  — ,    — ,     —,  respectively. 

Hence,  since  the  acceleration  of  motion  in  any  direction 
is  measured  by  the  rate  of  change  of  the  velocity  in  that 
direction,  we  have  for  the  accelerations  parallel  to  the  axes 
of  coordinates  the  expressions 


or 


(7) 


Total  Acceleration.  17 

where,  in  accordance  with  Newton's  notation,  x,  y,  z  denote 
the  accelerations  parallel  to  the  axes  of  x,  y,  z,  respectively. 
The  total  acceleration  of  the  motion  of  the  point  is  the 
resultant  of  these  accelerations. 

It  is  plain  that  this  acceleration  is  independent  of  any 
previously  existing  velocity,  which  may  or  may  not  be  in  the 
same  direction. 

The  question  of  acceleration  in  curvilinear  motion  can 
also  be  treated  in  another  manner,  as  follows : — 

24.  Curvilinear  Motion,  Change  of  Velocity, 
Total  Acceleration.  — Suppose  a  point  to  move  in  a 
curvilinear  path,  and  from  any  point  0  let  the  line  OA 
be  drawn,  representing  in  magnitude  and  direction  the 
velocity  of  the  moving  point  at  any  c  B 
instant.  Let  OB,  in  like  manner,  /  ^^^^/ 
represent   its   velocity   at   the   end       /  ^^-^^ 

of  the   interval  of  time  At.     Join     ^^— ' 

AB,  and  complete  the  parallelo- 
gram OABC.  Then  the  velocity  represented  by  OB  is  equi- 
valent to  the  component  velocities  represented  by  OA  and 
OC;  but  if  the  velocity  of  the  point  had  not  changed  during 
the  interval  At,  it  would  have  been  represented  by  OA ;  hence 
OC,  or  AB,  represents  in  magnitude  and  direction  the  change  of 
velocity  in  the  time  At. 

Again,  since  the  acceleration  of  the  velocity  of  a  mov- 
able, at  any  instant  is,  in  all  cases,  measured  by  the  rate  of 
change  of  the  velocity  for  that  instant,  it  follows,  as  in  (5),  that 
if  we  regard  the  interval  of  time  At  as  becoming  infinitely 
small,  the  acceleration  of  the  motion  is  represented  by  the 

AB 

limiting  value  of  — — .     This  limiting  value  is  called  the  total 
At 

acceleration  of  the  motion  of  the  particle  at  the  instant. 

25.  Tangential     and     Normal     Accelerations. — 

Again,  suppose  a  to  denote  the  position  of  the  moving  point 
at  the  end  of  the  time  t,  and  b  its  position  after  a  small 
interval  of  time,  At,  and  draw  tangents  to  the  path  at  the 
points  a  and  b.  Also,  as  before,  from  any  point  0  draw  OA, 
OB  parallel  to  these  tangents,  and  representing  the  velocities 


18  Acceleration. 

at  a  and  b,  respectively.     Then,  by  the  preceding  Article, 
AB  represents  the  total  change 
in  the  velocity  in  the  interval 
At. 

Draw  AN  perpendicular  to 
OB,  and  suppose  the  velocity 
AB  resolved  into  the  two,  AN 
and  BN;  then,  the  former  re- 
presents the  resulting  change  of 
velocity  in  the  normal  direction, 
and  the  latter  in  the  tangential. 

The  corresponding  accelerations  are  represented  by  the 

AN         BN 
limiting  values  of  — —  and  — ,  respectively. 

Again,  let  the  angle  BOA,  or  the  angle  between  the 
tangents  at  a  and  b,  when  indefinitely  small,  be  denoted  by 
d(p,  and  we  have 

AN  =  OAdcj>  =  vd<p. 

The  normal  acceleration  is  therefore 

dt         ds  dt         ds      P   K    M  '  /»     w 

where  p  represents  the  radius  of  curvature  of  the  path  at  the 
point  a. 

Also  in  the  limit  we  have  — -r  =  —.      Hence  the  tan- 

At       dt 

gential  acceleration  is  represented  by  —  ;   as  is  also  easily 

seen  from  equation  (6). 

In  the  case  of  uniform  motion  in  a  circle,  since  the  velo- 
city v  is  constant,  the  tangential  acceleration  vanishes,  and 
the  normal  acceleration  (which  then  becomes  the  total  accele- 

05  4  JJ    JJ 

ration)  is-,  or  -=^-,  where  r  denotes  the  radius  of  the  circle 

and  T  the  time  in  which  the  circle  is  described. 

The  normal  acceleration  in  this  case  is  called  the  centri- 


Hodograph,  19 

2)dal  acceleration,  as  it  is  constantly   directed  towards   the 
centre  of  the  circle. 

26.  Hodograph.* — In  accordance  with  the  method  of 
the  preceding  Articles,  if  from  any  point  0  lines  OA,  OB, 
OC,  &c,  be  drawn  representing,  in  magnitude  and  direction, 
the  velocities  at  the  points  at  b,  c,  &c,  taken  consecutively  in 
the  path  of  a  particle,  then  the  system  of  points  A,B,C,  &c, 
will  lie  on  a  new  curve  called  the  hodograph  of  the  original 
curve,  which  is  considered  to  be  described  by  the  point  A  as 
a  moves  along  the  given  curve. 

Since  the  lines  AB,  BC,  &c,  become  ultimately  tangents 
to  the  hodograph,  it  follows  that  the  direction  of  the  total 
acceleration  at  any  point  a  is  parallel  to  the  tangent  to  the 
hodograph  at  the  corresponding  point  A. 

Also,  since  the  total   acceleration   is  measured   by   the 
AB 
limiting  value  of  — — ,  it  follows  that  the  total  acceleration, 

at  any  point  a,  is  represented  by  the  velocity  at  the  point  A  in 
the  hodograph. 

We  shall  give  some  applications  of  this  method  subse- 
quently, more  especially  in  connexion  with  the  treatment  of 
Central  Forces. 

27.  Angular  Velocity,  Angular  Acceleration. — If 
the  position  of  a  point  P  moving  in  a  plane  be  taken  in  polar 
coordinates,  r  and  0,  with  reference  to  a  fixed  origin  0,  then 
the  rate  of  increase  of  the  angle  6  is  called  the  angular  velo- 
city of  P  relative  to  the  fixed  point  0.  Hence,  if  w  denote 
the  angnlar  velocity  at  any  instant, 

we  have  w  =  —  =  9. 
at 

Again,  if  P  move  along   OA, 

its  velocity  is  represented  by  —  ;  if 

a  0 

rJQ 

it  move  perpendicular  to  OA,  its  velocity  is  r  — ,   or  ruj. 

at 


*  Sir  W.  R.  Hamilton,  to  whom  this  method  is  due,  emplo3Ted  this  name 
(dtibv  ypdcpeiv)  (Proceedings,  R.  I.  A.,  1846,  p.  344)  in  his  discussion  of  the 
connexion  between  acceleration  and  motion.  The  hodograph  is  called  the 
curve  of  accelerations  by  French  writers  on  Mechanics. 

c2 


20  Acceleration. 

Hence  we  easily  see  that  the  most  general  motion  in  the 

dr 

plane  is  one  componnded  of  a  radial  velocity  — ,  along  with 

at 

a  perpendicular  velocity  r  — . 

If  OP  revolve  uniformly,  completing  its  revolution  in  T 
seconds,  then  its  angular  velocity,  in  circular  measure,  is 
obviously  given  by  the  equation 

•  =  ~-  (9) 

Suppose  OA  taken  equal  to  the  unit  of  length,  then  the 
velocity  of  the  point  A,  in  its  circular  path,  represents  the 
angular  velocity  of  the  line  OP. 

Again,  if  the  angular  velocity  of  P  be  variable,  its  rate  of 
increase  is  called  its  angular  acceleration  ;  hence  the  angular 

acceleration  of  P  with  regard  to  0  is  represented  by  —  or  — . 

at       at 

If  x  and  y  be  the  coordinates  of  P,  we  have 
x  =  r  cos  0,     y  =  r  sin  6  ; 
consequently,  when  r  is  constant,  we  get 

x  =  -  rta  sin  0  =  -  toy  \ 

y  =     rw  cos  0  =      u)X  J  ' 

These  give  the  components  of  velocity  of  any  point  which 
moves  in  a  circle,  in  terms  of  the  coordinates  and  the  angular 
velocity. 

28.  Accelerations  along  and  perpendicular  to  the 
Radius  Vector. — Let  x,  y  be  the  rectangular  coordinates 

of  the  moving  point  P,  and  r,  6  the  corresponding  polar 
coordinates,  at  the  end  of  the  time  t ;  then  x  and  y  (Art. 
23)  represent  the  accelerations  parallel  to  the  axes ;  hence, 
by  Art.  22,  the  acceleration,  P,  along  the  radius  vector  is 

/»     »   .    /k        %x  +  yy 

x  cos  0  +  y  sin  0,  or ; 


Area!  Velocity.  21 

and  the  acceleration,  T,  perpendicular  to  the  radius  vector  is 

u  cos  u  -  x  sm  u  = -. 

j  r 

To  find  expressions  for  these  accelerations  in  terms  of  r 
and  9,  we  have 

xx  +  yy  -  rr ; 

hence  xx  +  yy  +  xr  +  if  =  rr  +  f2 ; 

but  x2  +  if  =  r  +  r262 ; 

accordingly,  xx  +  yy  +  rO2  =  rr ; 

xx  +  in)      ..        ;.„ 

therefore  —  =  r  -  r6\ 

r 

Also  x'ij  -yx  =  —  (xy  -  yx)  =  -  (r20). 

Consequently,  the  acceleration  along  the  radius  vector  is 

*-'-  +  •><?-'[&):  (11) 

And  that  perpendicular  to  the  radius  vector  is 

If  the  acceleration  of  the  moving  particle  be  always 
directed  to  the  fixed  point   0,  we  have  T  =  0,  and  hence 

•>•'  _  =  constant ;  from  which  we  infer  that  the  radius  vector 

cit 
describes  equal  areas  in  equal  times  round  the  point  0. 

Equations  (11)  and  (12)  above  can  otherwise  be  obtained 
with  great  facility  by  a  method  analogous  to  that  employed 
in  Art.  25. 

29.    Areal    Velocity,     Areal    Acceleration.  — It   is 

obvious,  geometrically,  that  rdd  represent  double  the  area 


22 


Acceleration. 


described   by   the  line   OP  in  the  time   dt ;    consequently 
—r—  represents  the  rate  of  increase  of  double  the  area  de- 

2  fjfi 

scribed  by  the  point  P  round  the  point  0.     Hence  -£-  —r— 

ctt 

is  called  the  areal  velocity  of  the  point  P  relative  to  the 
origin  0.     Similarly  \  —  (  r2  —  )  represents  the  areal  accele- 


dt  \     dt  t 
ration  of  P  relative  to  the  same  origin. 

30.  Moving  Axes. — In  some  cases  it  is  necessary  to- 
refer  the  motion  of  a  point  in  a  y 
plane  to  rectangular  axes,  which 
are  themselves  in  motion.  Thus 
let  OX,  OF  be  two  fixed  rect- 
angular axes  in  the  plane,  and 
OM,  ON  be  two  moving  axes. 

Let  P  be  any  point  in  the  plane ; 
then  £  =  OM,  rj  =  ON,  where  t  and  r) 
are  the  coordinates  of  P,  relative  to  the  moviDg  axes. 

rid 

Also,  if  9  =  L  XOM,  we  have  w  =  — ,  the  angular  velocity 

at 

of  the  moving  axes.     Then  the  motion  of  P  is  got  by  com- 
pounding the  motions  of  M  and  N. 


dl 


Now,  by  Art.  27,  the  components  of  the  velocity  of  M  are 


—   along  OM,  and  w£  along  MP.    Likewise,  the  components 

CITi 

for  N  are  —  along  ON,  and  -  wy]  along  NP. 
at 

Hence,  if  u  and  v  denote  the  components  of  the  velocity 

of  P,  relative  to  the  moving  axes,  we  have 


dl 

dt 


I)bJ 


dx) 


(13) 


Again,  by  Art.  28,  the  acceleration  of  M  along  OM  is- 


Units  of  Time  and  Space.  23 

-~  -  w%  and  that  along  MP  is  p  —  (w£2)  ;  with  similar  ex- 

at  s  wt 

pressions  for  the  accelerations  of  N. 
Hence,  finally,  we  get 

d2£  1  d 

acceleration  parallel  to  0M=  —  -  w2£  -  -  -r.  (wrj2) ; 

(14) 
acceleration  parallel  to  ON  =  —  -  w2r]  +  ^-r  (w£2) • 

31.  Units  of  Time  and  Space. — With  respect  to  the 
units  of  time  and  space,  as  well  as  of  all  other  quantities,  it 
should  he  remarked  that  the  units  assumed  must  in  all  cases 
he  finite  magnitudes.  For  instance,  the  unit  of  time  may  be 
taken  as  a  second,  an  hour,  a  day,  or  any  other  finite  interval 
of  time,  but  it  should  never  be  assumed  to  be  an  indefinitely 
small  portion  of  time  ;  for  if  so,  numbers  which  represent 
finite  intervals  of  time  become  infinitely  great,  and  accord- 
ingly arguments  based  on  such  an  assumption  become  illusory 
and  unmeaning  when  applied  to  finite  intervals  of  time. 
This  remark  is  requisite,  as  fallacious  proofs  are  sometimes 
given  in  books  on  dynamics  from  overlooking  this  obvious 
principle. 

The  unit  of  time  most  universally  adopted  is  a  second,  as 
already  stated.  Different  units  of  length  prevail  in  different 
countries.  Since  in  this  country  the  foot  is  the  standard  of 
length,  and  areas  and  volumes  are  each  referred  to  units  of 
their  own,  we  shall  sometimes  employ  such  units  for  the 
purpose  of  illustrating  mechanical  principles  by  familiar 
examples.  But,  when  desirable,  we  shall  avail  ourselves  of 
the  metric  system.  In  it  the  unit  of  length  is  a  metre 
(3-2809  feet,  or  39-37079  inches).  From  this,  by  the  simple 
processes  of  squaring  and  cubing,  units  of  area  and  volume 
are  derived ;  and  decimal  multiples  and  submultiples  are 
respectively  indicated  by  the  use  of  Greek  and  Latin  pre- 
fixes. For  example,  the  centimetre  is  the  hundredth  part 
of  the  length  of  a  metre.  Again,  one  cubic  decimetre  is 
the  measure  of  capacity  called  a  litre,  and  is  about  61  cubic 
inches,  or  1-76  pints.  We  shall  subsequently  see  that  a 
cubic  centimetre  of  distilled  water  at  its  greatest  density 


24  Acceleration. 

furnishes  this  system  with  another  unit :  to  this  the  name 
gramme  is  applied.  One  thousand  grammes  are  called  a 
kilogramme,  equivalent  to  about  two  and  one-fifth  pounds 
avoirdupois. 

It  should  also  be  observed  that  in  the  numerical  expres- 
sion for  an  acceleration  there  is  a  double  reference  to  the 
unit  of  time;  so  that,  in  strict  accuracy,  what  we  have  called 
an  acceleration  of  7  feet  per  second  should  be  called  an 
acceleration  of  7  feet  per  second  per  second.  This  mode  of 
expression  is,  however,  cumbrous,  and  quite  unnecessary, 
since  in  ordinary  language,  as  well  as  in  mathematical  de- 
ductions, it  is  assumed  that  velocities  and  their  rates  of 
change  are  referred  to  the  same  unit  of  time,  unless  the 
contrary  be  stated. 


First  Law  of  Motion.  25 


CHAPTEE    III. 

LAWS    OF    MOTION. 

Section  I. — Rectilinear  Motion. 

32.  Motion  in  relation  to  Force. — In  the  preceding 
Chapters  motion  has  been  considered  from  a  purely  kine- 
matical  point  of  view ;  we  now  proceed  to  consider  it  in 
connexion  with  the  force  or  forces  by  which  it  is  produced. 

The  science  of  Rational  Dynamics  is  usually  founded  on 
three  principles,  or  Laws  of  Motion,  which  have  been  stated 
in  their  simplest  form  by  Newton,  and  are  fully  verified  by 
their  agreement  with  experience.  In  the  present  Chapter  it 
is  proposed  to  discuss  and  illustrate  various  cases  of  applica- 
tion of  these  Laws,  chiefly  when  the  forces  supposed  to  act 
are  constant  both  in  direction  and  magnitude.  The  discus- 
sion of  motion  produced  by  varying  force  will  be  dealt  with 
subsequently.  We  follow  Newton's  method,  commencing 
with  the  statement  of  his  First  Law. 

33.  First  law  of  Motion. — A  body  continues  in  its 
state  of  rest,  or  of  straight  uniform  motion,  except  in  so  far  as  it 
is  compelled  to  alter  that  state  by  impressed  force. 

This  law  asserts  that  a  body  has  no  power  or  tendency  in 
itself  to  alter  either  its  velocity  or  the  direction  of  its  motion  : 
this  is  usually  called  the  Law  of  Inertia  of  Matter. 

Hence,  if  a  body  be  conceived  to  be  set  in  motion,  and  no 
external  force  act  upon  it  afterwards,  it  should  continue  to 
move  indefinitely  in  a  right  line  with  a  uniform  velocity. 

Conversely,  if  a  body  be  in  a  state  of  uniform  rectilinear 
motion,  we  infer  that  the  forces  which  act  on  it  are  in  equili- 
brium. For  example,  if  a  train  be  in  a  state  of  uniform  motion 
on  a  horizontal  railway,  we  infer  that  the  force  arising  from 
the  action  of  the  steam  is  exactly  equal,  and  opposite  to,  the 
entire  resistance  arising  from  friction  and  resistance  of  the 
air. 


26  Rectilinear  Motion. 

Hence,  all  questions  of  uniform  rectilinear  motion  may  be 
regarded  as  problems  of  equilibrium,  and  treated  by  the  principles 
arrived  at  in  Statics.  In  all  applications  of  the  Laws  of  Motion 
to  a  body  of  finite  dimensions,  the  only  motion  considered  in 
this  Chapter  is  one  of  pure  translation. 

Again,  if  the  motion  of  a  body  be  not  uniform,  or  not 
rectilinear,  we  infer  that  it  must  be  acted  on  by  some  ex- 
ternal force  or  forces.  The  connexion  between  the  motion 
produced  and  the  force  which  produces  it  is  contained  under 
the  next  Law. 

Example. 

A  railway  train  is  moving  with  constant  velocity  along  a  horizontal  rail- 
road. The  resistance  from  friction,  &c,  for  each  carnage  is  one-hundredth 
part  of  the  pressure.  Find  the  tension  of  the  couplings  of  the  last  carriage,  if 
its  weight  he  four  tons. 

In  this  case,  since  the  motion  is  uniform,  the  tension  of  the  couplings  must 
be  equal  to  the  resistance  to  be  overcome,  or  to  the  one-hundredth  part  of  four 
tons,  i.e.  89f  lbs. 

34.  Second  Iiaw  of  Motion. — Change  of  motion*  is 
proportional  to  the  impressed  motive  force,  and  takes  place  in  the 
right  line  in  which  that  force  is  impressed. 

As  this  statement  is  very  comprehensive,  it  will  be^  neces- 
sary to  dwell  on  it  with  some  detail,  commencing  with  the 
case  of  a  body  under  the  influence  of  a  force  which  acts  uni- 
formly and  in  the  same  right  line  during  the  motion.  The 
body  is  supposed,  in  the  first  instance,  to  start  from  rest,  and 
the  direction  of  the  force  to  pass  constantly  through  its  centre  of 
mass,  in  which  case  the  motion  is  one  of  translationf  solely. 


*  For  the  present  we  shall  consider  that  it  is  one  and  the  same  body  which  is 
acted  on  by  forces  passing  through  its  centre  of  mass,  in  which  case  the  force 
varies  directly  as  the  velocity  generated  in  the  unit  of  time.  We  shall  subse- 
quently treat  of  the  case  where  the  mass  acted  on  varies  also.  In  that  case,  by 
the  word  "motus,"  here  translated  motion,  we  must  understand  quantity  of 
motion. 

t  A  force  applied  at  the  centre  of  mass  of  a  rigid  body  is  equivalent  to 
an  indefinite  number  of  equal  and  parallel  forces  applied  to  the  several  equal 
particles  of  which  the  body  is  conceived  to  be  constituted  ;  but  as  the  forces  are 
equal,  and  the  masses  moved  by  each  are  equal,  the  velocities  generated,  in  the 
same  time,  are  also  equal :  hence  the  motion  of  the  entire  body  is  one  of  pure 
translation.  The  simplest  case  of  this  is  that  of  bodies  falling  under  the  action 
of  the  force  of  gravity. 


Velocity  Generated.  27 

35.  Velocity  Generated. — Suppose  a  force  to  act  uni- 
formly on  a  body,  and  let /denote  the  velocity  generated  at 
the  end  of  the  first  second  (taken  as  the  unit  of  time) ,  then 
during  the  next  second,  in  accordance  with  our  law,  the  uni- 
form force  will  generate  an  additional  velocity  of  the  same 
amount/;  and  in  each  successive  second  the  force  generates 
the  same  additional  velocity ;  consequently  the  motion  is  in 
this  case  uniformly  accelerated,  and  the  velocity  at  the  end  of 
t  seconds  (Art.  17)  is  given  by  the  equation 

v  -A 

Again,  if  the  body  be  supposed  to  start  with  the  velocity 
«70  in  the  direction  in  which  the  force  acts,  we  shall  have  for 
the  velocity  v,  at  the  end  of  the  time  t, 

V  =  V0  +ft,  (1) 

as  in  Art.  17. 

If  the  force  act  in  a  direction  opposite  to  that  of  the 
motion  it  is  called  a  retarding  force ;  which,  if  uniform,  will 
diminish  the  velocity  by  the  quantity  /  during  each  second, 
and  we  shall  have,  as  before,  the  equation 

V  =  r0  -ft. 

The  student  should  bear  in  mind  that  /  in  all  cases  is 
measured  by  the  velocity  generated  or  destroyed  in  the  movable 
in  each  second  during  the  motion ;  /consequently  may  always 
be  regarded  as  an  acceleration — a  retardation  being  considered 
as  a  negative  acceleration. 

It  may  be  observed  that  the  entire  reasoning  in  this 
Article  depends  on  the  following  principle — contained  in  the 
Second  Law  of  Motion — that  the  change  of  velocity  produced 
by  a  force  in  any  time  is  independent  of  the  previous  velocity  of 
the  movable. 

The  Second  Law  of  Motion  equally  applies  to  the  case  of 
a  body  acted  on  by  any  number  of  forces,  in  which  case  it  may 
be  stated  as  follows  :  — 

If  any  number  of  forces  act  simultaneously  on  a  body,  then, 
during  any  instant,  each  force  produces  the  same  change  of  motion 
in  its  own  direction  as  if  it  had  acted  singly  on  the  body. 

Prom  this  it  follows  that  forces  are  compounded  in  the 


28  Rectilinear  Motion. 

same  manner  as  velocities.  The  law  of  the  composition  of 
forces  was  thus  establishad  by  Newton — Leges  Mot  us,  Cor.  2. 

36.  Space  described  in  any  Time. — Since  we  have 
seen  that  in  the  case  of  a  uniform  force  the  velocity  is  uni- 
formly accelerated  or  retarded,  we  can  at  once  apply  the  re- 
sults already  arrived  at  in  Arts.  18,  19. 

Hence,  the  space  described  from  rest,  in  the  time  t,  is 
given  by  the  formula 

s  =  i/t\  (2) 

If  the  body  start  with  an  initial  velocity  v0  along  the 
line  in  which  the  force  acts,  we  shall  have 

8  =  V0t  ±  \ft\  (3) 

in  which  the  upper  or  lower  sign  is  taken  according  as  the 
uniform  force  acts  in  the  same  or  the  opposite  direction  to 
that  of  the  initial  velocity. 

It  is  plain  that  the  space  described  in  the  first  second 
from  rest  is  J/,  or  half  the  velocity  acquired  at  the  end  of  the 
second;  and,  in  general,  the  space  described  in  anytime  from 
rest  is  half  of  that  described  by  a  body  moving  uniformly 
with  the  velocity  acquired  at  the  end  of  the  time. 

37.  Relation  between  Velocity  and  Space  de- 
scribed.— If  the  body  start  from  rest,  by  eliminating  t 
between  the  equations  v  =ft  and  s  =  ^ft2,  we  get 

v-=2fs; 

and,  more  generally,  if  v0  be  the  initial  velocity, 

v*=v*±2fs.  (4) 

From  the  preceding  results  it  is  seen  that  the  question  of 
rectilinear  motion  under  the  action  of  a  constant  force  is  com- 
pletely solved  whenever  the  value  of  the  acceleration /can  be 
determined.  In  a  subsequent  Article  we  shall  show  how  this 
can  be  done  in  elementary  cases,  but  before  doing  so  we  pro- 
ceed to  apply  the  preceding  results  to  the  important  case  of 
falling  bodies. 

38.  Vertical  Motion. — In  order  to  get  rid  of  the  re- 
tardation caused  by  the  resistance  of  the  air,  we  shall  sup- 


Vertical  Motion.  29- 

pose  the  motion  to  take  place  in  a  vacuum.  Under  these 
circumstances  it  is  found  that  all  bodies,  no  matter  what 
their  density  or  chemical  constitution  may  be,  fall  through 
the  same  vertical  height  and  acquire  the  same  velocity  in  the 
same  time.  That  this  is  so  is  best  established  by  means  of 
pendulum  experiments  ;  but  it  can  also  be  tested  by  allowing 
different  bodies  to  fall  in  an  exhausted  receiver.  We  hence 
infer  that  the  attractive  force  of  the  Earth  acts  equally  on 
all  bodies. 

If  g  denote  the  acceleration  due  to  the  force  of  gravity, 
that  is  the  increment  of  velocity  per  second  acquired  by  a  body 
falling  in  a  'vacuum,  then,  from  what  has  been  stated,  the 
value  of  g  is  the  same  for  all  bodies  at  the  same  place  on  the 
Earth's  surface. 

Again,  since  at  any  place  the  force  of  gravity  may  be 
assumed  as  a  constant  force  (t.  e.  within  moderate  distances 
from  the  Earth's  surface),  we  may  apply  to  the  case  of  falling 
bodies  the  results  arrived  at  in  the  preceding  Articles  by  sub- 
stituting g  in  place  of/.  Hence,  if  the  body  start  from  rest, 
we  have 

•  =  9t>     8=  \gt\     e  =  2gs.  (5) 

Again,  if  it  start  downwards  with  a  given  vertical  velo- 
city 0„, 

v  =  Vo  +  gt,     s  =  v0t  +  igt2,     v2  =  ev8  +  2gs.  (6) 

If  the  body  be  projected  vertically  upwards  with  a  velocity 
Vo,  gravity  becomes  a  uniformly  retarding  force,  and  we  have 

v  =  Vo-  gt,     s  =  vf  -  igt2,     v2  =  v02  -  2gs.  (7) 

To  find  in  this  case  the  height  H  to  which  the  body 
would  ascend,  we  make  v  =  0  in  the  last  equation,  and  we  get 

*-t  (8) 

The  time  T  of  ascent  is  given  in  like  manner  by  the 
equation 

r=-°.  (9) 

9 

The  subsequent  motion  of  the  body  is  got  from  equations 
(5),  in  which  we  suppose  the  body  to  start  from  rest  at  the 


30  Rectilinear  Motion. 

height  H.  It  immediately  follows  that  the  times  of  ascent 
and  descent  are  equal,  and  that  the  body  returns  to  its  ori- 
ginal position  with  the  velocity  with  which  it  was  projected 
upwards.  For  this  reason  we  say  that  the  velocity  v0  is  due  to 
the  height  H;  and  reciprocally,  that  the  height  H  is  due  to 
the  velocity  v0.  We  shall  meet  frequent  applications  of  these 
expressions. 

As  the  motion  is  supposed  to  take  place  in  a  vacuum,  the 
preceding  results  can  only  be  regarded  as  approximate  for 
motion  in  the  air. 

39.  Variation  of  Gravity. — It  is  found  that  the  value 
of  g  varies,  within  small  limits,  from  place  to  place  on  the 
Earth's  surface.  It  increases  with  the  latitude,  and  when 
referred  to  feet  and  seconds,  has  its  least  value,  32*091,  at 
the  equator,  and  its  greatest,  32*255,  at  the  pole.  It  also 
diminishes  as  the  body  is  raised  above  the  Earth's  surface, 
since  the  attraction  of  the  Earth  varies  as  the  inverse  square 
of  the  distance  from  its  centre.  The  value  of  g  at  London, 
referred  to  the  same  units,  is  32*19,  and  this  may  be  em- 
ployed, in  ordinary  calculations,  as  an  average  value. 

It  will  be  seen  subsequently  that  the  rotation  of  the 
Earth  on  its  axis  has  the  effect  of  diminishing  the  velocity  of 
a  falling  body ;  and,  accordingly,  the  observed  value  of  g  is 
the  difference  between  its  value  arising  from  the  Earth's 
attraction  and  the  component  of  the  centrifugal  acceleration 
in  the  vertical  direction. 

As  a  rough  approximation  we  may  assume  g  =  32 ;  and, 
when  numerical  results  are  required,  this  may  be  taken  as  its 
value  in  these  and  all  subsequent  examples,  unless  otherwise 
specified,  inasmuch  as  they  are  given  chiefly  for  the  purpose 
of  familiarizing  the  student  with  the  application  of  mechani- 
cal principles. 

Examples. 

1.  Find  the  velocity  acquired  in  5  minutes  by  a  falling  body,  assuming 
ff  =  32-19.  Arts.  9657  feet. 

2.  In  what  time  will  a  falling  body  acquire  a  velocity  of  400  feet  per  second 
if  it  start  from  rest  ?  Ans.  12-5  sec. 

3.  If  a  body  move  under  the  action  of  a  constant  force,  its  average  velocity 
during  anytime  is  an  arithmetical  mean  between  its  velocities  at  the  commence- 
ment and  the  end  of  that  time  ? 


Examples.  31 

4.  If  one  minute  be  taken  as  the  unit  of  time,  what  should  be  taken  as  the 
value  of  g  ? 

Ans.  The  velocity  per  minute  acquired  in  one  minute  by  a  falling  body,  or 
115,200  feet. 

5.  Two  bodies  start  together  from  rest,  and  move  in  directions  at  right  angles 
to  each  other.  One  moves  uniformly  with  a  velocity  of  3  feet  per  second ;  the 
other  moves  under  the  action  of  a  constant  force :  determine  the  acceleration 
due  to  this  force  if  the  bodies  at  the  end  of  4  seconds  be  20  feet  apart. 

Ans.  2  feet  per  second. 

6.  If  a  uniform  force  generate  in  a  body  a  velocity  of  30  feet  a  second  after 
•describing  25  yards,  find  the  acceleration.  Ans.  f=  6. 

7.  A  stone  is  let  fall  from  a  height  into  a  well,  and  is  heard  to  strike  the 
water  after  t  seconds  ;  find  the  depth  of  the  well;  assuming  the  velocity  of  sound 

to  be  V,  and  neglecting  the  resistance  of  the  air.  ^/S 

The  required  height  h  is  got  by  solving  the  equation 


~\  9 


h 

In  applying  this  equation  practically,  it  may  be  observed  that  —  is,  in  all  cases, 

small  in  comparison  with  t :  accordingly,  if  we  transpose  and  square,  we  get, 
neglecting  —  in  comparison  with  — , 


2(V+gt) 

8.  A  person  drops  a  stone  into  a  well,  and  after  three  seconds  hears  it  strike        ^/ 
the  water.     If  the  velocity  of  sound  be  1127  feet  per  second,  find  the  depth  of 

the  water.  Ans.  132-68  feet. 

9.  Prove  that  the  spaces  described  by  a  falling  body  in  successive  equal        y 
intervals  of  time  are  proportional  to  the  series  of  odd  numbers. 

10.  A  body  moves  from  rest  under  the  action  of  a  constant  force  during  four 
seconds,  when  the  force  is  supposed  to  cease  ;  in  the  next  five  seconds  the  body 
describes  200  feet;  find  the  acceleration  due  to  the  constant  force — (1)  if  one 
second  ;  (2)  if  one  minute  be  taken  as  the  unit  of  time. 

Ans.  (1)  10  ;  (2)  36000. 

11.  A  body  is  projected  upwards  with  any  velocity,  and  t,  t'  denote  the 
times  in  which  it  is  respectively  above  and  below  the  middle  point  of  its  path ;      v  ■< 

find  the  value  of  |.  Ans'  ^2+1' 

12.  Assuming  g  to  be  represented  by  32  when  the  units  of  space  and  time  are 
one  foot  and  one  second;  what  number  would  represent  its  value  if  one  mile  and 
one  day  be  taken  as  the  units  ?  Ans.  45242181&. 

" «  13.  A  ball  is  dropped  from  the  masthead  of  a  ship  sailing  n  miles  an  hour. 
Through  how  many  feet  must  it  have  fallen  when  the  direction  of  its  motion  is 
inclined  at  45°  to  the  horizon  ?  .       121  "2 

*"*'   3600* 


32  Rectilinear  Motion. 

40.  Acceleration  Varies  as  Pressure. — If  we  sup- 
pose different  forces  to  act  uniformly  during  equal  times  on 
the  same  body,  it  follows  from  the  Second  Law  of  Motion 
that  the  forces  will  be  to  one  another  in  the  same  ratio  as  the 
velocities  generated  in  equal  times. 

If  we  suppose  the  time  of  action  to  be  one  second,  the 
velocities  generated  are  represented  by  the  corresponding 
accelerations  /  and  f.  Also,  if  F,  F'  denote  the  statical* 
measures  of  the  forces,  i.  e.  the  total  pressures  which  they 
are  capable  of  producing,  we  have 

F\F'=f:f.  (10) 

If  one  of  the  constant  forces  be  the  attraction  of  the 
Earth,  since  its  statical  measure  is  W,  or  the  weight  of  the 
body  moved ;  and  since  g  is  the  corresponding  acceleration, 
we  have 

F:W=f:g;  (11) 

F 

hence  ^=W0'  ^ 

This  equation  enables  us  to  determine  the  velocity  generated 
in  one  second  by  a  constant  force  at  any  place  whenever  the 
pressure  F  which  measures  the  force  is  known,  and  also  the 
weight  of  the  body.  We  suppose,  as  stated  already,  that  the 
body  is  rigid,  and  that  the  force  F  acts  through  its  centre  of 
mass.  When  /has  been  determined  by  the  foregoing  equa- 
tion, and  the  force  continues  to  act  uniformly,  we  may  apply 
the  results  arrived  at  in  the  preceding  Articles  to  determine 
the  subsequent  motion  (see  Art.  37) . 

41.  Mass. — Our  ordinary  experience  suggests  to  us  that 
the  amount  of  the  acceleration  produced  in  a  body  by  a  force 
depends  not  only  on  the  magnitude  of  the  force  but  also  on 
the  body  which  is  moved.  When  exact  experiments  are 
carried  out  it  is  found  that  the  same  force  acting  on  different 


*  The  magnitude  of  a  force  is  estimated  in  Statics  by  the  weight  which  it 
is  just  capable  of  supporting.  Thus,  a  force  which  is  capable  of  supporting  a 
weight  of  112  lbs.  is  called  a  force  of  112  lbs.,  &c. 


Mass.  33 

bodies  produces  different  accelerations,  and  that  different 
forces  acting  on  the  same  body  produce  accelerations  pro- 
portional to  the  forces.  Hence  we  conclude  that  the  accelera- 
tion produced  in  the  motion  of  a  body  by  a  force  is  equal  to 
that  force  multiplied  by  a  factor  which  is  invariable  for  the 
same  body,  but  which  varies  for  different  bodies. 

Conversely,  if  F  denote  the  magnitude  of  a  force,  and  / 
that  of  the  acceleration  thereby  produced,  we  have  the  equa- 
tion 

F=  mfy  (13) 

where  m  is  always  the  same  for  the  same  body,  but  varies  for 
different  bodies.  This  quantity  m  is  called  the  Mass  of  the 
body,  and  is  estimated,  like  other  quantities,  by  comparing  it 
with  a  standard  quantity  of  the  same  kind.  It  is  found  that 
at  any  fixed  place  on  the  Earth's  surface  the  weight  of  a 
body  (if  permitted  to  accelerate  its  motion)  produces  an  ac- 
celeration which  is  the  same  for  all  bodies  (Art,  38).  Now 
W  being  the  weight  and  g  the  acceleration  thereby  produced, 
we  have  as  above  W-mg\  but  g  is  the  same  for  all  bodies 
at  the  same  place,  hence  W  is  proportional  to  m ;  or,  in  other 
words,  if  there  be  two  bodies  whose  weights  are  W,  W,  and 

whose  masses  are  m.  m',  we  have  ==,  =  — .      Hence,  in  order 

W      m 

to  find  the  ratio  of  the  masses  of  two  bodies,  we  have  only  to 

find  the  ratio  of  their  weights  at  the  same  place. 


Examples. 

1.  A  uniform  pressure  of  6  lbs.  is  applied  in  a  horizontal  direction  to  a  body 
of  10  lbs.  mass  placed  on  a  smooth  horizontal  table.  Find — (1)  the  velocity  gene- 
rated in  one  second ;  (2)  that  acquired  after  describing  500  yards  along  the 
plane.  <Ans.  (1)  19£;  (2)  240. 

2.  If  a  uniform  pressure  of  3  lbs.  produce  a  velocity  of  10  feet  in  the  first 
second,  find  the  weight  of  the  body  acted  on.  Ans.  9-6  lbs.^ 

3.  Find  the  pressure  which,  acting  uniformly  during  one  second,  will  gene- 
rate in  a  body  of  one  ton  mass  a  velocity  of  10  miles  per  hour. 

Ans.  9  cwt.  18f  lbs.  pressure. 

4.  If  a  pressure  of  one  ounce  act  uniformly  on  a  body  of  one  pound  mass, 
find  the  velocity  generated  from  rest  in  one  minute.  Ans.  ^g. 

D 


34  Rectilinear  Motion. 

5.  If  a  uniform  force  generate  in  a  mass  of  10  lbs.  a  velocity  of  30  feet  after 
describing  25  yards,  find  the  statical  measure  of  the  force.  60 

^itlS*    • —  IDS* 

9 

6.  A  pressure  F,  acting  on  a  mass  M,  generates  in  it  a  velocity  V,  in  the 

time  T;  find  the  value  of  F.  V 

'  Ans.  F=M—. 

7.  A  train  of  100  tons  acquires  on  a  horizontal  railroad  in  four  minutes  a 
velocity  of  30  miles  an  hour  ;  find  the  statical  measure  of  the  excess  of  the 
moving  above  the  retarding  pressure,  each  being  assumed  to  be  uniform. 

Arts.  11  cwt.  1  qr.  23J  lbs. 

8.  A  train  of  60  tons  is  impelled  along  a  horizontal  road  by  a  constant 
pressure  of  720  lbs.  Supposing  it  to  start  from  rest,  find  its  velocity  at  the 
end  of  one  minute — (1)  neglecting  friction ;  (2)  assuming  the  resistance  of  friction, 
air,  &c,  to  be  8  lbs.  per  ton.  Ans.  (1)  fog  ;  (2)  fog. 

9.  If  a  uniform  force  of  6  lbs.  produce  in  a  second  a  velocity  of  0*634  feet 
in  a  body,  express  the  quantity  of  matter  in  the  body  in  terms  of  cubic  feet  of 
water,  assuming  the  weight  of  a  cubic  foot  of  water  to  be  62|  lbs.  and 
^  =  32-19.  Ans.  4-87. 

10.  A  mass  of  450  lbs.  is  placed  on  a  perfectly  smooth  table:  a  uniform 
horizontal  pressure  is  exerted  on  it  which  increases  its  velocity  3  feet  in  every 
second ;  find  the  magnitude  of  the  pressure  in  lbs. 

Ans.  41  lbs.,  assuming  #  =  32-19. 

42.  Motion  on  a  Smooth  Inclined  Plane. — Let  us 

suppose  a  body,  starling  from  rest,  to  slide  under  the  influence 
of  gravity  down  a  perfectly  smooth  inclined  plane.  Let  i  de- 
note the  inclination  of  the  plane  to  the  horizon,  and  Wthe 
weight  of  the  body.  Eesolve  JFinto  its  components,  JFsin  i 
acting  parallel  to  the  plane,  and  W  cos  i  perpendicular  to  the 
plane.  The  motion  down  the  plane  is  evidently  due  to  the 
former  component,  since  the  latter  only  causes  pressure  on  the 
plane. 

As  the  force  along  the  plane  is  constant  and  acts  in  the 
direction  of  motion,  we  get,  substituting  Wsin  i  for  F  in  (12), 

f=gemi.  (14) 

Hence,  if  g  sin  i  be  substituted  for  /  in  the  f ormulce  in 
Arts.  36  and  37,  we  get 

v  =  gt  sin  i,     s  =  \gtf  sin  t,     v2  =  2gs  sin  i.  (1 5) 

We  assume  that  the  body  slides  without  rolling  along  the 


Motion  on  a  Smooth  Inclined  Plane. 


35 


plane,  as  otherwise  the  motion  would  not  be  one  of  pure  trans- 
lation. 

43.  Velocity  acquired  in  Moving  down  an  Inclined 
Plane. — Let  /  represent  AB,  the  length  A 

of  the  plane,  and  h  its  height  AC;  then 
if  v  be  the  velocity  acquired  on  arriving 
at  B,  we  have 


=  2gl  sin  i  =  2gh. 


(16) 


Accordingly,  the  velocity  acquired  at  any  point  in  the  descent  of 
a  body  down  a  smooth  inclined  plane  is  that  due  to  the  vertical 
height  through  which  the  body  has  descended.  This  is  a 
particular  case  of  an  important  principle  which  shall  be 
subsequently  considered. 

44.  Time  of  Descending  a  Chord  of  a  Vertical 
Circle. — We  next  proceed  to  show  that  the  time  of  de- 
scent down  any  chord  of  a  vertical  circle, 
starting  from  its  highest  point,  is  constant. 

Let  AC  be  the  vertical  diameter  of 
the  circle,  AB  any  chord  drawn  from  A. 

AB 

Join  BC;  then,  sine  =  sin  BCA  =  — -^; 

A  L> 

and,  if  T  be  the  time  of  descent  for  AB, 

we  have,  by  (15), 


AB 


A  Ti 
iaT2——       •    T2-2 
AC    ' 


hence 


w/ 


(17) 


where  a  denotes  the  radius  of  the  circle. 

Hence,  the  time  down  any  chord  such  as  AB  of  this  circle 
is  constant.  It  can  at  once  be  seen,  in  like  manner,  that  the 
time  of  descent  down  BC  has  the  same  value. 

45.  Line  of  Quickest  Descent  to  a  Circle. — To  find 
the  right  line  down  which  a  particle  under  the  action  of 
gravity  would  descend  in  the  shortest  time  from  a  given 
point  0  to  a  given  vertical  circle. 

d2 


36 


Rectilinear  Motion. 


Draw  AC,  the  vertical  diameter  of  the  circle,  and  join 
OC,  meeting  the  circle  in  B,  then  OB 
is  the  line  of  quickest  descent  in 
question.  For,  join  AB,  and  pro- 
duce it  to  meet  the  vertical  drawn 
through  0  in  D.  Then  it  is  ohvious 
that  the  circle  described  on  OD  as 
diameter  touches  the  given  circle  in 
B ;  consequently  the  time  of  descent 
down  OB  being  the  same  as  that 
down  any  other  chord  of  the  circle 
OBD,  drawn  from  0,  is  less  than  the  time  down  any  other 
right  line  drawn  from  0  to  meet  the  circle  ABC. 

The  preceding  method  of  investigation  applies  equally  if 
the  point  0  lie  inside  the  given  circle. 

46.  Iiine  of  Quickest  or  Slowest  Descent  to  any 
Curve. — It  is  easily  seen  from  the  preceding  Article  that  the 
determination  of  the  right  line  of  quickest  or  slowest  descent 
to  any  given  vertical  curve  from  any  point  in  its  plane  re- 
duces to  the  problem  of  drawing  a  circle,  touching  the  given 
curve  and  having  the  given  point  for  its  highest  point. 

The  problem  admits  also  of  being  treated  by  the  ordinary 
method  of  maxima  and  minima,  as  follows  : 

Suppose  the  curve  referred  to  polar  co- 
ordinates, the  given  point  0  being  taken  as 
pole,  and  the  vertical  OD  through  it  as 
prime  vector ;  then,  if  t  be  the  time  of  de- 
scent down  any  radius  vector  OP,  we  have 


r-igfcosO,  out 


■h 


2r 


cos  6 


Accordingly,  the  time  t  is  a  maximum  or  a  minimum  when 
r     . 


COS0 


is  a  maximum  or  a  minimum. 


u  = 


To   find    the    maximum   or  minimum    values,    assume 

„  ;  then  since  — .  =  0,  we  have 

cos  0  cW 


cos  0  — ,  +  r  sin  0  =  0. 
dU 


(18) 


Line  of  Quickest  Descent.  37 

The  solutions  are  obtained  by  combining  this  equation 
with  that  of  the  curve. 

To  distinguish  between  the  maximum  and  minimum 
solutions,  we  proceed  to  differentiate  the  equation 

cos  9  -TTj  +  r  sin  0 
die  ad 


eld  cos20 

dr 
observing  that,  in  this  case,  cos  9  —„  +  r  sin  9  =  0.   Hence  [Biff. 

Calc,  Art.  138),  t  is  a  minimum  or  a  maximum  according  as 

r  +  —  is  positive  or  negative. 

These  results  can  be  readily  verified  from  geometrical 
considerations. 

Examples. 

1.  If  the  hypothenuse  of  a  right-angled  triangle  be  placed  in  a  vertical 
position,  prove  that  the  times  of  descending  from  rest  will  be  the  same  for  each, 
of  its  sides. 

2.  Prove  that  the  velocity  acquired  down  any  chord,  terminated  at  the  lowest 
point  of  a  vertical  circle,  is  proportional  to  the  length  of  the  chord. 

3.  If  the  length  of  an  inclined  plane  be  150  yards,  and  its  inclination  30*, 
what  velocity  would  a  body  acquire  in  descending  it  ? 

Am.  40  yards  per  second. 

4.  A  body  slides  down  a  smooth  inclined  plane  of  given  height;  prove  that 
the  time  of  descent  varies  as  the  length  of  the  plane. 

5.  Find  the  inclination  of  a  plane,  of  given  length  I,  so  that  the  velocity 
acquired  in  moving  down  it  shall  be  of  a  given  amount  V.  .    .  _  V2 

JX'tlS.    Sill  t  —   T      La 

6.  Given  the  base  a  of  an  inclined  plane,  find  its  height  so  that  the  hori- 
zontal velocity  acquired  by  descending  it  may  be  the  greatest  possible. 

Ans.  h=a. 

7.  Find  the  gradient  in  a  railway  so  that  a  carriage  descending  the  plane  by 
its  own  "weight  may  move  through  one  quarter  of  a  mile  in  the  first  minute ;  and 
find  how  far  the  carriage  will  move  in  the  next  minute,  friction  being  neglected. 

(1)  sin  t=  4^0;     (2)  f  of  a  mile. 

8.  A  body  is  attached  by  a  string  to  a  point  in  a  smooth  inclined  plane,  on 

which  it  rests  :  if  it  be  projected  from  its  position  of  rest  up  the  plane  with  a 

velocity  just  sufficient  to  take  it  to  the  highest  point  to  which  the  string  allows 

it  to  go,  find  the  time  of  its  motion.         I     j 

Ans.  t  =  2    / — : — ,  the  length  of  the  string  being  /. 
yjff  sin . 


y 


/ 


38  Rectilinear  Motion. 

9.  A  groove  is  cut  in  an  inclined  plane,  making  an  angle  a  with  the  inter- 
section of  the  plane  and  the  horizon.  If  a  heavy  particle  he  allowed  to  descend 
the  groove  (supposed  smooth) ,  prove  that  its  acceleration  is  g  sin  i  sin  o ;  where 
i  denotes  the  inclination  of  the  plane. 

10.  If  two  vertical  circles  have  a  common  highest  point,  then  if  any  line  he 
drawn  from  that  point,  the  time  of  descending  the  portion  intercepted  between 
the  circles  is  constant. 

11.  Find  the  right  line  of  quickest  descent  from  a  point  to  a  given  right 
line  lying  in  the  same  vertical  plane  as  the  point. 

12.  Find  the  right  line  of  quickest  descent  from  a  given  right  line  to  a  given 
vertical  circle. 

13.  Find  the  lines  of  quickest  and  slowest  descent  between  two  vertical 
circles  which  lie  in  the  same  plane. 

14.  A  parabola  whose  latus  rectum  is  p  is  placed  in  a  vertical  plane,  with  its 
axis  horizontal.  Find  the  inclination  of  the  normal  terminated  by  the  axis  down 
which  a  particle  would  descend  in  the  shortest  time,  and  find  the  time  of  its 
descent.  •  j^Z 

Am.  i  =  45°,  time ; 


9     ■ 

15.  Find  the  latus  rectum  of  a  parabola,  so  that  when  it  is  placed  in  a  ver- 
tical plane  with  its  axis  horizontal  the  least  time  in  which  a  particle  falls  from 
rest  down  a  normal  from  the  curve  to  the  axis  may  be  one  second. 

16.  Prove  that  the  chords  of  quickest  and  slowest  descent  from  the  highest 
or  to  the  lowest  point  of  a  vertical  ellipse  are  at  right  angles  to  each  other,  and 
parallel  to  the  axis  of  the  curve. 

17.  Show  immediately,  from  equation  (18),  that  the  right  line  of  quickest 
descent  from  a  given  point  to  a  given  curve  makes  equal  angles  with  the  nor- 
mal at  its  extremity  and  the  vertical ;  and  verify  the  result  geometrically. 

18.  An  ellipse  is  placed  with  its  major  axis  vertical ;  find  the  semi-diameter 
along  which  a  particle  will  descend  in  the  shortest  time  possible  from  the  cir- 
cumference to  the  centre. 

Am.  It  makes  with  the  axis  major  the  angle  sec-^V^),  where  e  is  the 
eccentricity.     If  e  <  — ,  the  line  of  quickest  descent  is  the  axis  major. 

19.  An  ellipse  is  placed  with  its  major  axis  vertical ;  find  the  line  of  quickest 
descent  from  the  upper  focus  to  the  curve. 

Am.  It  makes  with  the  axis  major  the  angle  cos"1  — .     If  e<|,the  axis 

major  is  the  required  line. 

20.  AB  is  a  quadrant  of  a  circle  whose  centre  is  0,  the  radius  OB 
being  horizontal;  C  is  a  point  on  the  quadrant,  and  the  angle  BOG—d. 
Show  that  the  time  of  falling  from  A  to  C  is  to  that  of  falling  from  C  to  B  as 


Jcos?toJ^f. 


Parabolic  Motion.  39 


Section  II. — Parabolic  Motion; 

47.  Path  of  a  Projectile. — We  have  hitherto  considered 
rectilinear  motion  ;  we  now  proceed  to  the  case  of  a  body 
projected  in  any  direction,  and  acted  on  only  by  the  force  of 
gravity,  which  is  supposed  to  be  uniform. 

In  this  case  it  is  easily  shown  that  the  path*  described  by 
the  projectile  is  a  parabola. 

For,  suppose  a  body  projected  from  0  with  the  velocity  V, 
in  the  direction  OX,  and  draw  OY  xt    ^x 

vertically  downwards. 

Let  ON  he  the  space  which  the 
body,  moving  with  the  velocity  V, 
would  describe  in  t  seconds ;  then, 
if  no  force  were  to  act  on  the  body,  m 
N  would  represent  its  position  at 
the  end  of  that  time. 

Again,  as  the  force  of  gravity  acts  in  the  direction  OY,  it 
will  produce  its  effect  in  that  direction,  by  the  Second  Law  of 
Motion,  independently  of  the  previous  velocity  of  the  body : 
i.  e.  it  will  produce  the  same  effect  as  if  the  body  fell  freely 
from  rest.  Measure  off,  accordingly,  OM=\gf ;  then  OM 
represents  the  space  moved  through  in  the  vertical  direction 
in  the  time  t. 

Complete  the  parallelogram  OMPJSf,  and  by  the  combined 
effect  of  the  two  motions  P  will  be  the  position  of  the  projec- 
tile at  the  end  of  the  time  t. 

Let  xy  y  be  the  co-ordinates  of  P  referred  to  the  axes 
OX  and  OY,  and  we  have 

x  =  ON=Vt,    y=OM=±gt\ 

If  t  be  eliminated  between  these  equations,  the  equation  of 
the  path  described  is  Q  ~2 

*•-—*.  (i) 


*  As  before,  by  the  path  described  by  a  body  we  understand  the  path  de- 
scribed by  its  centre  of  mass. 


40  Parabolic  Motion. 

This  equation  represents  a  parabola,  touching  OX  and  having 
its  axis  vertical. 

If  H  be  the  height  due  to  the  velocity  V  (Art.  38),  the 
equation  of  the  parabola  becomes 

x*  =  ±Hy.  (2) 

48.  Construction  for  Focus  and  Directrix. — From 
the  preceding  equation  it  follows  (Salmon's 
Conic  Sections,  Art.  214)  that  H  is  the  dis- 
tance of  0  from  the  focus  of  the  parabola, 
and  also  from  its  directrix. 

Hence,  if  OD  be  measured  vertically 
upwards  equal  to  IT,  and  DR  drawn  inQ 
a  horizontal  direction,  the  line  DR  will  be 
the  directrix  of  the  parabolic  trajectory. 

Also  if  OF  be  drawn  through  0,  making  the  angle  XOF 
equal  to  the  angle  XOD,  and  if  we  take  OF  =  OB ;  then  F 
will  be  the  focus  of  the  trajectory. 

Hence,  as  the  focus  and  directrix  of  the  curve  are  known, 
it  is  completely  determined. 

Again,  the  velocity  at  any  point  in  the  trajectory  is  equal 
to  that  which  the  body  would  acquire  in  falling  from  the  direc- 
trix. 

We  have  seen  that  this  property  holds  good  for  the  point 
of  projection :  moreover,  after  passing  through  any  point  the 
body  will  move  in  the  same  path  as  if  it  had  been  projected 
from  that  point,  in  the  direction  and  with  the  velocity  that 
it  has  at  the  instant;  therefore  the  property  is  true  for  any 
point  in  the  path. 

Hence,  whenever  the  velocity  at  any  point  is  given,  the 
position  of  the  directrix  is  completely  determined. 

Definition. — The  angle  which  the  direction  of  projection 
makes  with  the  horizontal  line  is  called  the  angle  of  elevation 
of  the  projectile. 

49.  Horizontal   Range  and  Time  of  Flight. — Let 

R  be  the  point  in  which  the  projectile  strikes  the  horizontal 
plane  through  0;  then  OR  is  called  the  horizontal  range, 


Range  and  Time  of  Flight  for  an  Oblique  Plane.         41 


and  the  time  Tof  describing  the  corresponding  path  is  called 
the  time  of  flight. 

Through  R  draw  RQ  in  the 
vertical  direction. 

Let  OR  =  R,  L  QOR  =  e ;  then 
we  have 

OQ  =  FT,  QR  =  \gT\ 
But  QR  =  OQ  sin  e  ;  hence  we  get 

2  Fsin  e 


T  = 


(3) 


V2  . 
Also  R  =  OQ  cos  e  =  FT  cos  e  =  2  —  sin  e  cos  e 


therefore 


R  =  2Hsm2e. 


(4) 


If  F  be  given,  the  horizontal  range  is  the  greatest  when 
sin2<?  =  1,  or  e  =  45°. 

The  maximum  horizontal  range  is  accordingly  25",  or 
double  the  height  due  to  the  velocity  of  projection. 

50.  Range  and  Time  of  Flight  for  an  Oblique 
Plane. — First   suppose   it  an   ascend-  .q 

ing  plane,  and  let  %  be  its  inclination, 
and  e  the  angle  of  elevation  QON. 
Then,  as  before,  we  have 

OQ  =  FT,     QR  =  \gT\ 

But  in  the  triangle  QOR,  we  have     o 


hence 


or 


QR      sin  (e  -  i) 

OQ 

cosi 

sin  (e  - 

0    gT 

cos  i 

2V 

2Fsin(*-0 

(5) 


cos* 


42  Parabolic  Motion. 

Also  the  range 


therefore 


COS  I  COS  I 


_     2  V2  sm  (e  -  i)  cos  e  ,ox 

R  = t: .  (6) 

a  cos~* 


In  the  case  of  a  descending  plane  it  is  easily  seen  that 
the  range  and  time  of  flight  are  obtained  by  changing  the 
sign  of  i  in  the  preceding  results. 

For  given  values  of  V  and  »,  R  becomes  a  maximum  when 
sin  [e  -  i)  cos  e  is  a  maximum,  or  when 

sin  (2e  -  i)  -  sin  i  is  a  maximum  ; 

but  this  is  greatest  when 

2*-t  =  90°,     or     e  =  i(90°  +  «). 

Hence,  the  direction  of  elevation  for  a  maximum  range 
bisects  the  angle  between  the  vertical  and  the  inclined  plane. 

Again,  since  in  this  case  OR  =  RQ,  the  maximum  range 
and  the  corresponding  time  of  flight  are  connected  by  the 
relation 

R=\gT\ 

From  the  value  of  e  found  above,  it  follows  immediately 
that  the  focus  of  the  parabola,  in  this  case,  lies  on  the  in- 
clined plane. 

51.  Given  the  velocity  of  projection  to  find  the  elevation  in 
order  to  strike  a  given  object. — Here,  in  formula  (6),  we  are 
given  V,  R,  and  t,  to  find  e.  Hence,  sin  [e  -  i)  cos  e  is  given, 
and  therefore  sin  [2e  -  i)  is  given,  from  which  e  can  be  de- 
termined. 

The  problem  admits  of  a  simple  geometrical  investigation 
also,  as  follows : — 

Let  0  be  the  point  of  projection,  and  P  the  position  of 
the  given  object.  Then,  since  the  velocity  of  projection  is 
given,  the  position  of  the  directrix  HK  is  known. 


Trajectory  referred  to  Vertical  and  Horizontal  Axes.      43 


Hence,  with  0  and  P  as  centres,  describe  circles  touching 
the  directrix,  and  let  F,  F'  be  their 
points  of  intersection.  These  points 
are  obviously  the  foci  of  the  two  pa- 
rabolic trajectories  which  satisfy  the 
proposed  conditions.  Hence  the  pro- 
blem admits  in  general  of  two  solu- 
tions. 

The  corresponding  directions  of  projection  are  found  by 
bisecting  the  angles  FOR  and  F'OH,  as  is  obvious  from  the 
elementary  properties  of  the  parabola. 

The  problem  becomes  impossible  when  the  circles  do  not 
intersect. 

The  range  in  the  direction  OP  is  obviously  a  maximum 
when  the  circles  touch  one  another.  In  this  case  there  is  but 
one  solution,  and  the  focus  of  the  parabola  lies  in  the  line  OP, 
as  already  seen. 

52.  Trajectory  referred  to  Vertical  and  Horizon- 
tal Axes.— Suppose  OX  and  OY  to  be  the  horizontal  and 
vertical  lines  drawn  through  the 
point  of  projection  0,  and  let  x,  y 
be  the  coordinates  of  P,  the  posi- 
tion of  the  projectile  at  the  end  of 
any  time  t. 

Let  OQ  be  the  direction  of 
projection,  and  resolve  the  initial 
velocity  V  into  its  horizontal  com- 
ponent, V  cos  e,  and  its  vertical,  V  sin  e.  Then,  since  the 
force  of  gravity  has  no  effect  on  the  horizontal  motion,  the 
component  Fcos  e  remains  constant  during  the  motion  ;  con- 
sequently we  have 

x  =  ON  =  Vt  cos  e. 

Also,  for  the  motion  in  the  vertical  direction,  we  get 
(Art.  38), 

y  =  Vt  sin  e  -  igt2 ; 


therefore 


y 


x  tan  e  - 


get? 


2  V2  cosre 


=  x  tan  e  - 


4JSTcos2e 


(7) 


44  Parabolic  Motion. 

This  equation  represents  a  parabola,  whose  axis  is  vertical, 
as  already  seen. 

Again,  if  v  be  the  velocity  at  the  point  P,  and  <p  the  angle 
the  direction  of  motion  makes  with  the  axis  of  x,  we  have 

v  cos  <j>  =  Fcos  e,     and     v  sin  <p  =  Fsin  e  -  gt; 

hence  #2  =  F2  cos2^  +  ( Fsin  e  -  ^^)2 

=  F2-2#(F?sine-  igt2) 

=  V*-2gy  =  2g(H-y). 

Hence,  as  already  shown  otherwise,  the  velocity  at  any 
point  is  that  acquired  by  a  body  falling  from  the  directrix. 

53.  Height  of  Ascent. — Since  vertical  and  horizontal 
motions  may  be  considered  separately,  it  follows  that  the 
greatest  height  above  the  horizontal  plane  is  that  to  which 
a  body  projected  vertically  with  the  velocity  V  sin  e  would 
ascend.     This  by  (Art.  38),  is 

F2sin2e  _  .  . 

— ,     or  Ji  sure. 

V  sin  b 

Also,  the  time  of  ascent  is ,  from  the  same  Article  : 

9 
a  result  which  can  also  be  obtained  by  finding  the  maxi- 
mum value  of  y  in  equation  (7).  From  these  the  same  ex- 
pressions as  before  for  the  range  and  the  time  of  flight  can 
be  easily  deduced  :  for,  the  whole  time  of  flight  is  obviously 
double  that  of  reaching  the  highest  point ;  and  the  range  is 
got  by  multiplying  the  value  so  found  by  V  cos  e. 

54.  I/PT,  FT  be  the  tangents  at  two  points  P,  F  on  a 
parabolic  trajectory,  and  vy  v   the  cor-  t 
responding  velocities,  to  prove  that                          //3\/3^^ 

v:v=PT:FT.  (8) 

The  line  joining  Tto  the  middle 
point  of  PF  is  vertical,  l 

being  parallel  to  the  axis  of  the  parabola.     Again,  let 

a  =  LTPF,    c[  =  lTFP,    $  =  lPTL,    (3'=lFTL. 


Projectiles. 


45 


Then,  since  the  horizontal  component  of  the  velocity  at  P 
is  equal  to  that  at  Pf ,  we  have 

v  sin  /3  =  v  sin  /3', 

£  _  sin^  _  H_ 
v~  sin  ]3  ~PT 


or 


Also,  since 


PT 
FT 


sin  a 
sin  a 


we  get 


v  sin  a  =  v  sin  a'.  (9) 

55.  Lemma. — If  6  he  the  angle  BD C  which  a  right  line 
CD  drawn  from  the  vertex  makes  with  the  base  of  a  triangle 
ABC,  we  have 

AB  cot  0  =  BD  cot  A-  AD  cot  B.  (10) 

For,  draw  OiV  perpendicular  to  AB,  and  we  have,  by 
elementary  geometry, 

AB.DN=AN.DB-AD.  BN.  c 

Hence,  dividing  by  CiV, 

._   DN    »„  AN     An  BN 
AB'CN=BDVN-AD'CN> 


or 


AB  .  cot  0  =  PZ>  cot  A-  AD  cot  P. 


Again,  if  a  and  ]3  be  the  angles        e 
which  CD  makes  with  AC  and  BC  respectively,  we  have 
AB  cot  9  =  AD  cot  a  -  PP  cot  j3. 

This  follows  at  once  by  drawing  AE  parallel  to  BC,  and 
applying  the  preceding  result. 

56.  Being  given  the  direction  and  the  velocity  of  projection, 
to  find  the  velocity  with  which  a  projectile  would  strike  an  oblique 
plane,  and  also  the  direction  of  its  motion  at  the  instant  of 
impact. 

Let  i  be  the  inclination  of  the  plane  to  the  horizon  ;  then, 
by  the  preceding  lemma  (see  figure  on  last  page], 

cot  a  -  cot  a  =  2  tan  i.  (11) 

Hence,  the  angle  a  is  determined  from  the  known  angles 
a  and  i. 


46  Parabolic  Motion. 

v  sin  o 


Again,   since      v  sin  a  =  v  sin  a',     we  have 
which  determines  v'. 


sin  a 


If  the  projectile  impinge  at  right  angles  on  the  plane,  we 
have  a  =  90° ;  therefore  cot  a  =  2  tan  t,  which  determines  a, 
or  the  corresponding  angle  of  elevation.  Also  the  velocity 
with  which  the  projecticle  strikes  the  plane  is  v  sin  a  in  this 
case. 

57.  motion  on  a  Smooth  Inclined  Plane. — In  our 

discussion  of  motion  on  an  inclined  plane  in  Art.  42  the 
movable  was  supposed  to  start  from  rest :  in  this  case  the 
motion  is  rectilinear.  It  is  also  rectilinear  if  the  initial 
motion  has  place  in  the  direction  of  the  line  of  greatest  slope 
in  the  plane.  But  when  the  body  is  projected  along  the  plane 
in  any  other  direction  the  problem  is  the  same  as  that  pre- 
viously discussed,  namely,  the  motion  of  a  projectile  acted  on 
by  a  constant  force,  parallel  to  a  given  direction.  Its  path 
along  the  plane  is,  accordingly,  a  parabola ;  and  its  axis  is 
in  the  direction  of  the  line  of  greatest  slope. 

58.  Morin's  Apparatus. — "We  conclude  with  a  short 
description  of  the  apparatus,  designed  by  Poncelet,  and  con- 
structed by  Morin,  for  experimentally  exhibiting  the  laws  of 
falling  bodies. 

A  cylinder  is  made  by  clock-work  mechanism  to  revolve 
around  a  fixed  vertical  axis.  A  weight  is  suspended  at  the 
summit  of  the  cylinder  close  to  the  outer  surface  and  between 
two  vertical  guides.  "When  the  rotation  has  become  perfectly 
uniform,  the  weight  is  allowed  to  fall.  A  pencil,  attached  to 
the  falling  weight,  is  so  arranged  as  to  trace  a  line  on  a  sheet 
of  paper,  which  is  wrapped  tightly  around  the  revolving 
cylinder.  When  the  paper  is  taken  off  and  unrolled  on  a 
plane  surface,  the  curve  traced  on  it  by  the  pencil  is  found 
to  be  a  parabola. 

That  this  curve  is  a  parabola,  may  be  shown  in  the  follow- 
ing manner : — 

Let  GP'P  represent  the  curve  traced  out  by  the  pencil. 


Monti's  Apparatus. 


47 


Draw  the  tangent  GL  to  the  curve  at  the  initial  point  G, 
and  at  any  point  P  draw  the  tangent  PL,  and  erect  LF 
perpendicular  to  it  at  the  point  L.  Make  a  corresponding 
construction  for  the  other  points  on  the  path ;  then  the  lines 
LF,  L'F,  &c,  are  all  found  to  intersect  in  a  common  point  F. 
This  is  a  characteristic  property  of  the  parabola  which  has  its 
focus  at  F,  and  its  vertex  at  G. 
Having  found  the  curve  to  be  a  para- 
bola, we  can  show  that  the  motion 
of  the  weight  has  been  uniformly 
accelerated.  Let  PM,  FN  be  the 
coordinates  of  P,  referred  to  the 
axes  GL,  GF,  then  if  t  denote  the 
time  in  which  the  moving  weight 
arrived  at  the  position  P,  the  line 
PM  will  be  equal  to  the  arc  of  the 

point  on  the  circumference  of  the  cylinder  has  rotated  in  the 
time  t.  Let  V  denote  the  constant  velocity  of  any  point  on 
the  circumference  of  the  cylinder,  and  we  get  PM  =  Vt. 

Again,  from  the  property  of  the  parabola, 


circle  through  which  a 


Accordingly, 


PM2=4:FGxMGt 

PM2       V2 
MG=4FG  =  ±FGt2'> 


but  MG  is  the  space  through  which  the  weight  has  descended 
vertically  in  the  time  t ;  hence  the  spaces  described  by  the 
falling  body  vary  as  the  squares  of  the  times ;  its  motion 
consequently  is  uniformly  accelerated. 

Comparing  with  the  equation  s  =  J  gf,  we  get  g  =  ; 

that  is,  the  distance  of  the  focus  of  the  parabola  from  its 
summit  is  equal  to  the  height  due  to  the  velocity  of  a  point 
on  the  surface  of  the  rotating  cylinder. 

The  student  can  easily  prove  that  the  parabola  described 
is  the  same  as  that  of  a  body  projected  horizontally  from  a 
point  with  the  velocity  V. 


48  Parabolic  Motion. 

59.  In  the  preceding  investigations  we  have  neglected  the 
effects  of  the  resistance  of  the  air.  When  this  is  taken  into 
account  the  problem  becomes  one  of  great  uncertainty,  arising 
from  the  law  of  resistance  of  fluids  not  being  accurately  known, 
and  from  the  difficulties  still  remaining  in  the  integration  of 
the  equations  of  motion,  when  the  law  of  resistance  is  assumed. 
The  most  generally  received  theory  is  that  the  resistance  of 
fluids  is  proportional  to  the  square  of  the  relative  velocity  of 
the  fluid  and  the  movable.  When  the  resistance  of  the  air 
is  taken  into  account,  it  is  easily  shown  that  the  preceding 
results  are  not  even  approximate  in  cases  of  high  velocity ; 
such,  for  instance,  as  shot  and  shell  projected  by  artillery. 

Examples. 

1.  Determine  the  elevation  of  a  projectile,  so  that  its  horizontal  range  may 
be  equal  to  the  space  to  be  fallen  through  to  acquire  the  velocity  of  projection. 

Am.  e  =  l5°. 

2.  If  a  number  of  particles  be  projected  simultaneously  from  the  same  point 
with  a  common  velocity,  but  in  different  directions,  prove  that  at  any  subse- 
quent instant  they  will  all  be  situated  on  the  surface  of  a  sphere. 

3.  Given  the  horizontal  range  and  the  time  of  flight  of  a  projectile  ;  find  its 
initial  velocity  and  angle  of  elevation. 

4.  If  a  body  be  projected  obliquely  on  a  smooth  inclined  plane,  the  path  in 
which  it  moves  will  be  a  parabola.  Find  the  position  of  the  focus  and  directrix 
of  the  parabola  when  the  initial  velocity  and  direction  of  motion  are  given. 

5.  Given  the  velocity  with  which  a  shot  is  projected  from  a  certain  point ; 
find  the  locus  of  the  extremities  of  the  maximum  ranges  on  inclined  planes  pass- 
ing through  that  point. 

6.  If  a  body  be  projected  with  a  velocity  of  100  feet  per  second  from  a  height 
of  66  feet  above  the  ground,  in  a  direction  making  an  angle  of  30°  with  the 
horizon  ;  find  when  and  where  it  will  strike  the  ground. 

Am.  Time  =  4£  sec.     Range  =  357*23  feet. 

7.  If  A,  B  be  two  points  on  a  parabolic  trajectory,  prove  that  the  time  of 
passage  from  one  to  the  other  is  proportional  to  tan  <p  —  tan  <p' ;  where  <p,  <p' 
represent  the  inclinations  to  the  horizon  of  the  tangents  drawn  at  A  and  B. 

8.  Given  the  initial  velocity,  find  the  angle  of  elevation  that  a  projectile 
should  just  clear  a  wall  at  a  given  distance  from  the  point  of  projection.  Find 
also  the  distance  at  which  the  body  strikes  the  ground  afterwards. 

9.  A  piece  of  ordnance,  under  proof  at  "Woolwich,  at  a  distance  of  50  yards 
from  a  wall  14  feet  high,  burst,  and  a  fragment  of  it,  originally  in  contact  with 
the  ground,  after  just  grazing  the  wall,  fell  6  feet  beyond  it  on  the  opposite 
side.     Find  how  high  it  rose  in  the  air. 


Examples.  49 

10.  When  the  velocity  of  projection  is  given,  all  the  parabolas  which  can  he 
described  in  the  same  plane  by  a  projectile  are  enveloped  by  a  fixed  parabola : 
prove  this,  and  hence  find  the  maximum  range  on  a  given  plane. 

7 

11.  A  body  is  projected  with  a  velocity  of  100  feet,  in  a  direction  inclined  at 
an  angle  of  60D  to  the  horizon  :  find  its  least  velocity  during  the  motion,  and  the 
time  of  attaining  it.  Ans.  50  feet ;  2-7  seconds. 

12.  If  two  bodies  be  projected  simultaneously,  with  a  common  velocity, 
from  the  same  point  on  an  oblique  plane,  one  upwards  and  the  other  downwards, 
and  if  the  directions  of  their  projection  make  equal  angles  with  the  inclined  plane, 
show  that  the  times  of  flight  are  equal.  The  motion  is  supposed  to  take  place 
in  a  plane  perpendicular  to  the  inclined  plane. 

13.  With  what  velocity  should  a  projectile  be  discharged  at  an  elevation  of 
•30°,  so  as  to  strike  an  object  at  a  distance  of  2500  feet  on  an  ascent  of  1  in  40  ? 

14.  Find  the  latus  rectum  of  the  parabola  described  by  a  projectile. 

The  velocity  of  tbe  highest  point  of  the  path  is  Fcos*,  but  it  is  also  equal  to 
the  velocity  acquired  in  falling  from  the  directrix  ;  therefore  the  latus  rectum 
.    2  V2 
is cos-£. 

9 

15.  If  a  body  be  projected  from  the  point  A  in  the  direction  of  AC,  and  from 
any  point  C  in  the  line  a  vertical  line  CD  be  drawn,  meeting  the  curve  described 
by  the  projectile  in  D  ;  again,  if  B,  the  middle  point  of  AC,  be  joined  to  D,  show 
that  BD  will  be  the  direction  of  the  motion  at  D,  and  that  the  velocity  at  D  will 
he  to  that  at  A  as  BD  is  to  AB. 

16.  A  number  of  bodies  slide  from  rest  down  the  chords  of  a  vertical  circle, 
starting  from  its  highest  point,  and  afterwards  move  freely :  prove  that  the  locus 
of  the  foci  of  their  paths  is  a  circle  whose  radius  is  half  that  of  the  given  circle. 

17.  If  bodies  be  projected  from  the  same  point  Math  velocities  proportional 
to  the  sines  of  their  elevations,  find  the  locus  of  points  arrived  at  in  a  given 
time.    * 

18.  Two  bodies  are  projected  simultaneously  in  different  directions  from  the 
same  point,  with  given  velocities  :  prove  that  the  line  which  connects  their  posi- 
tions at  each  instant  moves  parallel  to  a  given  direction. 

19.  Two  particles  are  projected  from  a  point  with  equal  velocities,  their 
directions  of  projection  being  in  the  same  vertical  plane — t,  t'  being  the  times 
taken  by  the  particles  to  reach  their  other  common  point,  and  T,  T'  the  times 
of  reaching  their  highest  points.  Show  that  tT  +  t'T  is  independent  of  the 
directions  of  projection. — Camb.  Trip.,  1876. 

20.  If  two  particles  be  describing  the  same  parabolic  trajectory,  prove  that 
the  right  line  connecting  them  envelopes  an  equal  parabola. — Ibid. 

21.  A  train  is  moving  at  the  rate  of  60  miles  an  hour  when  a  ball  is  dropped 
from  the  roof  inside  one  of  the  carriages.  Prove  that  the  ball  describes  a  para- 
bola in  space,  and  find  the  position  of  the  axis  and  directrix. 

If  the  height  of  the  carriage  be  9  feet,  and  the  ball  rebound  from  the  floor 
without  loss  of  velocity,  describe  by  means  of  a  figure  the  path  of  the  ball  in 
space  so  long  as  the  motion  continues. 

E 


50  Friction. 


Section  III. — Friction. 

60.  Iiaws  of  Dynamical  Friction. — Before  completing 
our  discussion  of  motion  under  the  action  of  a  constant  force, 
it  is  desirable  to  make  a  few  observations  on  the  resistance, 
arising  from  friction,  which  takes  place  when  one  body  slides 
on  another.  We  shall  consider  only  the  case  of  motion  along 
a  fixed  plane,  and  shall  assume  that  the  roughness  of  the  plane 
is  the  same  throughout.  Under  these  circumstances  the  laws 
of  friction — as  established  by  experiment — may  be  stated  as 
follows : — 

(1).  The  resistance  caused  by  friction  against  the  motion 
of  a  body  sliding  on  a  uniformly  rough  plane  is  proportional 
to  the  normal  pressure  which  the  body  exerts  against  the 
plane. 

(2).  It  is  independent  of  the  amount  of  surface  in  con- 
tact. 

(3).  It  is  independent  of  the  velocity  of  motion. 

(4).  The  ratio  of  the  friction,  during  the  motion^  to 
the  normal  pressure  is  called  the  coefficient  of  Dynamical 
friction. 

(5).  The  friction  between  two  substances  in  motion  is  in 
general  less  than  the  friction  in  the  state  bordering  on  motion, 
or  the  Statical  friction. 

(6).  The  mutual  friction  varies  with  the  nature  of  the 
surfaces  in  contact,  and  can  be  much  diminished  in  amount 
by  the  use  of  unguents,  as  also  by  polishing  the  surfaces  in 
contact. 

The  student  will  observe  that  the  laws  of  Dynamical  fric- 
tion are  in  every  respect  similar  to  those  of  Statical  friction 
(Minchin's  Statics,  Arts.  34-36). 

For  fuller  information  on  the  laws  of  Friction  the  student 
is  referred  to  Jellett's  Theory  of  Friction. 

61.  Motion  on  a  Rough  Horizontal  Plane. — Let  W 

be  the  weight  of  a  body  sliding  on  a  uniformly  rough  hori- 
zontal plane,  and  p.  the  relative  coefficient  of  friction ;  then, 


Motion  on  a  Bough  Inclined  Plane.  51 

since  in  this  case  the  normal  pressure  is  represented  by  W,  the 
friction  is  fxW;  and  since  it  acts  as  a  retarding  force  we  get 
by  Art.  40, 

f~-jpr0  =  W-  C1) 

Accordingly,  substituting  -  /ug  for  /  in  the  equations  of 
Arts.  35,  36,  and  37,  we  get 

v  =  V  -  fxgt 

v*=V*-2ms    L  (2) 

s  =rt-ifxgf  j 

By  means  of  these  equations  the  motion  is  completely 
determined  whenever  ji,  the  coefficient  of  friction,  and  V,  the 
initial  velocity,  are  known. 

To  find  when  the  body  is  brought  to  rest  by  the  friction, 
we  make  v  =  0  in  the  first  of  these  equations,  and  the  required 

V 

number  of  seconds  is  — .  Again,  the  space  moved  over  be- 
fore the  body  is  brought  to  rest  is  given  by  2/mgs  =  V2. 

62.  Motion  on  a  Rough  Inclined  Plane. — Suppose 
a  body  of  weight  W  to  slide  on  a  uniformly  rough  plane,  of 
inclination^;  then  resolving  W  into  its  components,  JFcos  i 
and  TFsin  i;  the  former,  IFcos  ?,  represents  the  pressure  on 
the  plane;  and  accordingly  the  friction  is  represented  by 
[xWcosi;  and  since  it  acts  against  the  motion,  we  have  for 
the  total  force  producing  motion  down  the  plane  the  expres- 
sion TFsin  i-fi  IF  cos  i.  If  this  value  be  substituted  for  F  in 
equation  (12),  Art.  40,  we  get 

f  =  g  (sin  i-  fx  cos  i).  (3) 

If  (p  be  the  limiting  angle  of  resistance  for  the  plane,  t.  e. 
if  ju  =  tan^,  the  preceding  formula  becomes 

sin  (i-  0) 


f-9 


COS0 


for  a  body  sliding  down  the  plane. 

The  corresponding  equations  connecting  velocity,  time, 

e2 


52  Friction. 

and  space,  are  had  by  substituting  this  value  for /in  the  for- 
mulae of  Arts.  36  and  37. 

If  the  body  be  projected  up  the  plane,  in  a  direction  at 
right  angles  to  the  intersection  of  the  plane  with  the  horizon, 
the  retarding  force  is  represented  by  IF  sin  i  +  fi  Wcosi : 
hence  the  value  of  /becomes 

/--0(sm*+/icose)  =  -0r— — --,  (4) 

when  we  introduce  for  n  its  value  tan  <£.     The  equations  con- 
necting s,  v9 1  can  be  found  immediately,  as  before. 


Examples. 

1.  A  body  is  projected  with,  a  velocity  of  100  feet  per  second  along  a  rough 
horizontal  plane  :  find,  assuming  /*=■&,  (1)  the  time  iu  which  it  is  brought  to 
rest  by  friction ;   (2)  the  whole  space  passed  over. 

Ans.  (1)  37£  seconds;  (2)  625  yards. 

2.  A  body  is  projected  with  a  velocity  of  100  yards  per  minute  along  a  rough 
horizontal  plane,  and  is  brought  to  rest  in  10  seconds :  find  the  coefficient  of  fric- 
tion. Am-  fJ-  =  -h- 

3.  A  train,  often  tons  weight,  is  impelled  by  steam  along  a  horizontal  railroad 
with  a  constant  pressure  of  630  lbs.    If  the  friction  be  7  lbs.  per  ton,  calculate— 

(1)  the  velocity,  in  miles  per  hour,  after  moving  from  rest  for  one  minute  ;  and 

(2)  the  space  described  in  that  time ;  neglecting  the  resistance  of  the  air,  &c. 

Ans.  (1)  32-A-  miles ;  (2)480  yards. 

(b)  If  the  steam  be  shut  off,  find  how  far  the  train  would  run 'before  it  is 
brought  to  rest  by  friction.  2  miles  320  yards. 

4.  A  body  projected  with  a  velocity  of  30  feet  is  brought  to  rest  after  sliding 
100  yards  on  a  rough  horizontal  plane  ;  find  the  coefficient  of  friction. 

Ans.  -6\. 

5.  A  body  is  projected  up  a  plane,  of  20  yards  length  and  30°  inclination, 
with  a  velocity  of  50  feet  per  second  :  find  the  coefficient  of  friction  that  the  body 
should  just  arrive  at  the  top  of  the  plane.  ^^       _    29 

96  V3* 

6.  Two  masses,  M,  M',  connected  by  a  string,  slide  down  a  rough  inclined 

plane  in  a  vertical  plane  at  right  angles  to  the  intersection  of  the  former  with 

the  horizon  :  if  the  coefficients  of  friction  be  fi  and  fx,  respectively,  prove  that 

,  •    .      -^"  +  ^V         •  > 
the  acceleration  down  the  plane  is^(sin» cos  i). 

7.  A  body  slides  down  a  rough  roof  and  afterwards  falls  to  the  ground  :  find 
the  whole  time  of  motion. 


Momentum.  53 

8.  Several  bodies  start  from  the  same  point  and  slide  down  different  inclined 
planes  of  the  same  roughness  :  find  the  locus  of  their  positions  after  the  lapse  of     .s 
a  given  time.     Find  also  the  locus  of  the  positions  arrived  at  with  a  common     ^ 
velocity.  "V"1  t-   , 

9.  A  rough  plane  makes  an  angle  of  453  with  the  horizon ;  a  groove  is  cut  in 
the  plane  making  an  angle  o  with  the  intersection  of  this  plane  and  the  horizon- 
tal plane  ;  if  a  heavy  particle  he  allowed  to  descend  the  groove  from  a  given       U^ 
height  h  find  the  velocity  with  which  it  reaches  the  horizontal  plane. 

Ans.     ffr*(«jn«-#Q. 

yf         sin  a 

10.  A  body  moves  from  rest  down  an  inclined  plane  whose  inclination  is  30°, 
and  limiting  angle  of  resistance  15°:  find  the  velocity  acquired  if  the  length  of 
the  plane  be  200  feet. 

Here  v2  =  400#  tan  15°;  therefore  v  =  58-56  feet  per  second. 

11.  A  railway  train  is  moving  up  an  incline  of  1  in  120  with  a  uniform 
velocity.  Find  the  tension  of  the  couplings  of  the  carriage  which  is  attached 
to  the  engine ;  assuming  the  weight  of  the  train  (exclusive  of  the  engine)  to  be 
80  tons,  and  the  friction  8  lbs.  per  ton. 

Ans.  19cwt.  5^  lbs. 

12.  In  the  same  case,  if  the  acceleration  of  the  train  be  2  feet  per  second, 
find  the  tension  of  the  couplings. 

/ 
Here  we  must  add  to  the  preceding  W  -,  i.  e.  5  tons  ;  and  the  entire  tension 

is  nearly  6  tons. 


Section  IV. — Moment 


urn. 


63.  Force  measured  by  Quantity  of  motion  gene- 
rated in  Unit  of  Time. — The  product  of  its  mass  and  the 
velocity  which  a  body  has  at  any  instant  is  called  its  quantity  of 
motion  or  momentum  at  that  instant.  Accordingly  we  con- 
clude, from  equation  (13),  Art.  41,  that  F  varies  as  the 
quantity  of  motion  it  can  generate  in  one  second  (taken  as  the 
unit  of  time),  the  force  being  supposed  to  act  uniformly 
during  that  time. 

Again,  since  the  velocity  (g)  which  gravity  can  produce 
in  one  second  is  the  same  for  all  bodies,  the  quantity  of 
motion  gravity  can  generate  in  one  second  in  a  falling  body 
of  mass  m  is  represented  by  mg ;  hence,  in  this  case,  we  have 

W  =  mg ; 

in  which  the  units  of  mass  and  weight  are  connected  in  such 
a  manner  that  when  one  is  fixed  the  other  is  also  determined. 


54  Momentum. 

64.  Absolute  Unit  of  Force. — In  accordance  with 
equation  (13),  Art.  41,  the  unit  of  force  is  defined  as  the  force 
which,  acting  uniformly  during  the  unit  of  time  on  a  unit  of  mass, 
produces  a  unit  of  velocity.  This  is  called  by  Gauss  the 
absolute  unit  of  force. 

The  most  convenient  unit  of  mass  in  the  British  Isles  is 
the  mass  contained  in  one  standard  pound  avoirdupois. 

Hence,  adopting  as  before  a  second  as  the  unit  of  time, 
and  a  foot  as  the  unit  of  length,  the  absolute  unit  of  force  is 
that  which,  acting  during  one  second,  would  produce  in  a 
standard  pound  mass  a  velocity  of  one  foot  per  second.  This 
unit  of  force  is  sometimes  called  a  poundal.  Hence,  if 
g  =  32*19  with  reference  to  the  preceding  units,  the  unit  of 
force  is  32T19  part  of  the  attraction  of  the  earth,  at  London,  on 
a  standard  pound ;  i.  e.  about  half  an  ounce,  approximately. 

In  the  metric  system  the  force  which  in  one  second  would 
generate  a  velocity  of  one  centimetre  per  second  in  a  gramme 
of  matter  is  called  a  dyne.  Hence,  since  1  lb.  =  453*6  grammes, 
and  1  foot  =  30*48  centimetres,  one  poundal  is  approximately 
13825  dynes. 

65.  Gravitation  Units  of  Force  and  Mass. — In 
practical  questions  concerning  bodies  on  the  earth's  surface, 
it  is  in  general  more  convenient  to  measure  forces  by  weights, 
and  to  speak  of  a  force  of  so  many  pounds  weight.  In  this 
system  the  unit  of  force  is  the  weight  at  some  definite  place 
(London)  of  the  pound  mass ;  or  of  a  kilogramme  when  the 
metric  system  is  taken.  This  is  called  the  gravitation  or 
statical  measure  of  force ;  and  since  the  unit  of  force  in  this 
system,  acting  on  one  pound  mass  for  one  second,  produces 
a  velocity  of  32*19  feet  per  second,  we  see  that  this  unit  is 
32*19  times  the  absolute  unit.  Moreover,  since  the  weight 
of  a  body  varies,  within  certain  small  limits  from  place  to 
place  (Art.  38),  when  scientific  accuracy  is  required  we  must 
correct  for  the  change  in  the  value  of  g  due  to  any  difference 
in  altitude  or  latitude  from  those  of  the  place  to  which  the 
standard  was  originally  referred. 

In  practice  this  correction  seldom  requires  to  be  taken 
into  account,  as  the  variation  in  the  value  of  g  is  generally 
too  small  to  aifect  the  result  appreciably  (Art.  39). 


Tuo  Classes  of  Forces.  55 


Examples. 

1.  An  ounce  being  taken  as  the  unit  of  mass,  a  second  as  the  unit  of  time, 
and  an  inch  as  the  unit  of  length,  compare  the  unit  of  force  with  the  weight  of 
one  pound. 

Here  the  unit  of  force  is  that  which  in  one  second  would  generate  a  velocity 

of  one  inch  per  second  in  an  ounce  mass ;  and  therefore  is  — — -—   pait 

of  the  weight  of  one  pound,  or  1*25  grains. 

2.  Determine  the  unit  of  time  in  order  that  g  may  he  expressed  by  unity 
when  the  foot  is  the  unit  of  length. 

Ans.  -  V2  seconds. 
8 

3.  Find  the  units  of  space  and  time  in  order  that  the  acceleration  of  a  body 
falling  in  vacuo,  and  the  velocity  it  acquires  in  one  minute,  may  respectively  be 
the  units  of  acceleration  and  of  velocity. 

66.  Two  Classes  of  Forces. — There  are  two  classes  of 
forces  to  be  considered  in  Dynamics  :  one,  such  as  gravity  and 
those  hitherto  discussed,  which  require  a  finite  time  to  pro- 
duce a  finite  change  of  velocity.  Forces  of  this  class,  when 
uniform,  are,  as  has  been  stated,  measured  by  the  change 
produced  in  one  second  (taken  as  unit  of  time)  in  the  mo- 
mentum of  the  body  acted  on.  There  is  another  class,  called 
ordinarily  impulses,  such  as  blows,  sudden  impacts.  &c,  which 
act  only  during  a  very  short  time,  but  are  capable  of  pro- 
ducing a  finite  change  of  velocity  in  that  time. 

These  are  sometimes  called  instantaneous  forces ;  it  is  ne- 
cessary, however,  to  observe  that  force  in  all  cases  requires 
some  time  to  produce  its  effects,  though  that  time  may  be 
exceedingly  small.  In  fact,  we  cannot  conceive  that  a  force 
could  produce  any  change  in  the  velocity  of  a  body  if  its 
time  of  action  were  absolutely  nothing. 

Forces  of  the  former  class  are  frequently  styled  finite  or 
continuous  forces,  to  distinguish  them  from  the  other  class, 
namely,  impulsive  forces. 

It  should  be  observed  that  whenever  both  impulsive  and 
finite  forces  act  at  the  same  time  on  a  body,  the  latter  may 
in  general  be  neglected  in  determining  the  motion  at  the 
instant;  since  the  effects  produced  by  them,  in  the  time 
during  which  the  impulsive  forces  act,  are  so  small  that  they 
may  be  neglected  in  comparison  with  the  effects  of  the  im- 
pulses. 


56  Momentum. 

67.  Impulses. — The  measure  of  an  impulse,  i.e.  of  the 
entire  action  of  a  force  of  great  intensity,  which  acts  during 
a  very  short  time,  and  then  ceases,  is  the  whole  change  in  the 
quantity  of  motion  which  it  communicates  to  the  body  on  which 
it  acts. 

We  may  here  observe  that,  if  .Fbe  the  instantaneous  value 
of  an  impulsive  force,  and  r  the  time  of  action,  the  whole 

impulse  is  represented  by  (  Fdt,  in  which,  as  already  observed, 

Jo 

t  is  a  very  small  interval  of  time. 

68.  General  Equations  of  Motion  of  a  Particle. — 

Suppose  that  the  force  F  acts  as  before  in  the  line  of  motion 
of  the  mass  acted  on,  but  that  it  varies  continuously,  then  we 
may  consider  that  in  an  indefinitely  small  portion  of  time  its 
intensity  is  unaltered.  The  variable  acceleration/,  caused  by 
it,  is  determined  by  the  equation  F=  mf:  hence,  as  in  Art.  21, 
we  have  at  any  instant 

dv        d-s         .. 
F  =  mf  =  m  —  =  m  -—  =  ms .  (6) 

at        dv 

Hitherto  the  motion  has  been  supposed  rectilinear.  In 
the  case  of  curvilinear  motion  the  last  equation  expresses  the 
tangential  component  of  the  force,  and  it  can  be  similarly 
seen  (Art.  25)  that  the  normal  component  is  expressed  by 

v2 
m  — .    We  now  proceed  to  consider  the  motion  of  a  particle 

P 
of  mass  m,  under  the  action  of  any  forces.  If  the  particle  be 
referred  to  a  system  of  rectangular  axes  in  space,  and  x,  y,  z, 
be  the  coordinates  of  its  position  at  any  instant,  i.  e.  at  the 
end  of  the  time  t,  reckoned  from  any  fixed  instant,  the  com- 
ponents of  its  velocity  parallel  to  the  axes  of  coordinates  are 
(Art.  12)  represented  by  x,  i/}  z. 

Resolve  the  whole  force  acting  on  the  particle  at  the 
instant  into  three  components,  parallel  to  the  axes  of  x,  y,  z, 
respectively ;  and  let  these  components  be  represented  by 
X,  Y,  Z;  then,  since  by  the  Second  Law  of  Motion  each 
of  these  forces  produces  its  change  of  velocity  in  its  own 


General  Equations  of  Motion  of  a  rartick. 


57 


direction,  we  deduce  from  what  precedes  (see  Art.  24)  the 
equations 

_  d  fdx\         d2x 

These  are  called  the  differential  equations  of  motion  of 
the  particle ;  and  the  solution  of  the  problem  depends  in  each 
case  on  the  integration  of  these  equations. 

As  already  stated,  the  preceding  equations  hold  for  the 
motion  of  any  rigid  body,  provided  the  direction  of  the  force 
which  acts  on  it  always  passes  through  its  centre  of  mass. 

69.  In  some  problems  the  mass  acted  on  constantly  varies 
during  the  motion  ;  in  this  case  equation  (3)  becomes 


F--(mv). 


(5) 


For  instance,  suppose  a  ball  projected  vertically  upwards, 
a  chain  of  indefinite  length  being  attached  to  it,  and  drawn 
up  gradually  by  it ;  to  investigate  the  motion. 

Here,  if  m  be  the  mass  of  the  ball,  fj.  that  of  a  unit  of 
length  of  the  chain,  and  s  the  length  of  chain  in  motion 
at  any  instant,  we  have  M  =  m  +  ps  ;  and  if  m  =  />>,  our  equa- 
tion gives 

ds\ 


dt\K 


dt\ 


or 


dt 


nf 


(k  +  s)g, 


n       \2   ds 


Hence 


{k+sy-(^j=c--y(k  +  sy. 


If  V  be  the  initial  velocity,  we  have 

&2F2=  c-yv. 


58  Action  and  Reaction, 

Hence 

This  determines  the  velocity  at  any  height ;  also  H,  the 
height  of  ascent,  is  given  by  the  equation 

s=«f^%-k-  (7) 

V2 
If  k  =  oo  ,  this  is  easily  seen  to  become  -jr— ,  which  agrees 

with  Art.  38. 

Section  V. — Action  and  Reaction. 

70.  Third  Law  of  Motion. — Reaction  is  always  equal 
and  opposite  to  action  :  that  is,  the  mutual  actions  of  two  bodies 
are  always  equal  and  take  place  in  opposite  directions. 

On  this  law  Newton  remarks  as  follows : — "  If  any  person 
press  a  stone  with  his  ringer,  his  finger  is  pressed  by  the  stone. 
If  a  horse  draw  a  body  by  means  of  a  rope,  the  horse  also  is 
drawn  (so  to  speak)  towards  the  body  ;  for  the  rope  being 
strained  equally  in  both  directions,  draws  the  horse  towards 
the  body  as  well  as  the  body  towards  the  horse,  and  impedes 
the  progress  of  one  as  much  as  it  promotes  that  of  the  other. 
Again,  if  any  body  impinge  on  another,  whatever  quantity  of 
motion  it  communicates  to  that  other  it  loses  itself  (on  account 
of  the  equality  of  the  mutual  pressure)." 

•    Newton  verified  this  law  experimentally  in  the  case  of  the 
collision  of  spherical  bodies. — See  Scholium,  Axiomata. 

He  also  showed  that  the  law  holds  good  in  the  case  of  the 
attraction  of  bodies,  as  follows  : — 

Let  A  and  B  be  two  mutually  attracting  bodies,  and  con- 
ceive some  obstacle  interposed  by  which  their  approach  to 
one  another  is  prevented.  If  the  body  A  be  acted  on  towards 
B  by  a  greater  force  than  B  is  acted  on  towards  A,  then  the 
obstacle  will  be  more  urged  by  the  pressure  of  A  than  by  the 
pressure  of  B.  The  stronger  pressure  should  prevail,  and 
cause  the  system  consisting  of  the  two  bodies  and  the  obstacle 
to  move  in  directum  towards  B ;  also,  as  the  force  is  uniform 
the  motion  would  be  accelerated  ad  infinitum,  which  is  absurd, 
and  contrary  to  the  first  law  of  motion ;  for,  by  that  law,  such 


Stress,  Forces  of  Inertia.  59 

a  system,  as  it  is  not  acted  on  by  any  external  force,  should 
continue  in  a  state  of  rest  or  of  uniform  rectilinear  motion. 

71.  Stress,  Forces  of  Inertia.— The  fact  is  that  force 
is  always  exhibited  as  a  mutual  action  between  two  bodies  ; 
and  this  phenomenon,  regarded  as  a  whole,  is  described 
by  the  term  stress,  of  which  action  and  reaction  are  but 
different  aspects.  Thus  to  the  action  of  a  force  producing 
an  acceleration  of  motion  in  a  body  corresponds  an  equal  and 
opposite  reaction  against  acceleration;  this  is  called  the  force 
of  inertia  of  the  body.  It  thus  folldfe  that  the  force  of  inertia 
of  any  material  particle  must  be  ^ual  and  opposite  to  the 
resultant  of  all  the  forces  which  act  on  the  particle,  whether 
arising  from  the  action  of  the  other  parts  of  the  system  or 
from  that  of  forces  external  to  the  system.  Hence,  inthe 
motion  of  any  material  system,  since  the  actions  and  reactions 
of  its  different  parts  equilibrate  in  pairs,  we  infer  that  there 
is  equilibrium  between  the  external  forces  which  act  on  the 
system  and  the  several  forces  of  inertia  of  the  different  par- 
ticles of  which  the  system  is  composed.  This  is  equivalent  to 
the  celebrated  principle  introduced  by  D'Alembert,  and  called 
by  his  name,  but  which  is  directly  implied  in  Newton's 
Scholium  on  the  Third  Law  of  Motion.  This  has  been  observed 
by  many  writers  on  Mechanics,  but  the  connexion  of  New- 
ton's Scholium  with  the  modern  theory  of  work  and  energy 
was  first  pointed  out  by  Thompson  and  Tait :  see  their  Treatise 
on  Natural  Philosophy,  vol.  i.,  pp.  247-8. 

72.  The  laws  of  Motion,  like  every  law  of  nature,  must 
ultimately  depend  for  their  establishment  on  their  agreement 
with  experiment  and  observation.  Accounts  of  the  different 
apparatus  that  have  been  devised  for  the  purpose  of  verify- 
ing these  laws  will  be  found  in  the  books  especially  devoted 
to  the  purpose,  such  as  Ball's  Experimental  Mechanics. 
The  most  complete  proof  of  the  laws  of  motion,  however, 
is  derived  from  Physical  Astronomy.  The  Lunar  motions, 
for  instance,  have  been  calculated  from  equations  depending 
solely  on  these  laws ;  and  the  observed  and  calculated  posi- 
tions are  found  to  agree  with  a  precision  that  could  only 
arise  from  the  perfect  accuracy  of  the  principles  from  which 
they  were  deduced. 


w' 


60  Action  and  Reaction. 

One  of  the  simplest  contrivances  for  illustrating  the  laws 
of  motion,  in  the  case  of  falling  bodies,  is  that  devised  by 
Atwood,  which  we  shall  now  proceed  to  consider. 

73.  Atwood's  Machine. — In  its  simplest 
form  this  machine  may  he  regarded  as  consisting 
of  two  masses  connected  by  a  string  which  passes 
over  a  small  fixed  pulley.  We  shall  neglect  the 
weight  of  the  pulley,  and  also  that  of  the  string, 
as  well  as  the  friction  at  the  axle  of  the  pulley. 

Suppose  W  and  W  |fc  represent  the  wreights 
of  the  bodies,  of  which  flRis  the  greater. 

Let  T  denote  the  tendon  of  the  string  at  any  instant : 
then  considering  the  pulley  as  perfectly  smooth,  this  tension, 
by  the  law  of  action  and  reaction,  must  act  equally,  and  in 
opposite  directions,  on  the  two  masses. 

Accordingly,  we  may  regard  the  body  W  as  acted  on  by 
the  pressure  W  downwards,  and  the  tension  T  upwards ;  i.  e. 
by  the  single  force  W  -  T  acting  downwards — then,  the 
corresponding  acceleration  /,  from  Art.  40,  is  given  by  the 
equation 

.     W-T 

Similarly,  the  upward  acceleration  of  the  other  body  is  repre- 

T  -  W 

sented  by      ^,     g. 

Again,  as  the  string  is  supposed  inextensible,  the  velocities 
of  the  bodies  at  any  instant  are  equal  and  opposite,  and  hence 
their  accelerations  also. 

Accordingly  we  have 

W-  T      T-  W 


w         w  ' 

or 

w+  w" 

a) 

This  determines  the  tension  of  the  string. 

Again, 

we  have 

W- 

9                            9 

At  wood's  Machine.  61 

therefore  W-W=(W+  W) '-, 

W-W 

or  /-  w+W0'  () 

This  determines  the  acceleration.  By  aid  of  it  the  velo- 
city and  the  space  described  in  any  time  can  be  readily 
deduced. 

The  most  important  advantage  of  this  apparatus  is  that, 
by  taking  bodies  of  nearly  equal  weights,  we  can  make  the 

acceleration  — — — ,  g  as  small  as  we  please. 

A  complete  account  of  Atwood's  apparatus  is  beyond  the 
scope  of  this  treatise.  In  a  subsequent  place  we  shall  consider 
the  modification  required  when  allowance  is  made  for  the  mass 
of  the  pulley. 

Examples. 

1.  A  mass  of  488  grammes  is  fastened  to  one  end  of  a  chord  which  passes 
over  a  smooth  pulley.     What  mass  must  he  attached  to  the  other  end  in  order      *f 
that  the  488  grammes  may  rise  through  a  height  of  200  centimetres  in  10  seconds, 
assuming^  =  980  centimetres?  Ans.  492  grammes. 

2.  Two  weights  of  14  and  18  ozs.  are  suspended  hy  a  fine  thread  which 
passes  over  a  smooth  pulley,  if  the  system  be  free  to  move ;  find  how  far  the       \r 
heavier  weight  will  descend  in  the  first  three  seconds  of  its  motion,  and  also  the 
tension  of  the  string.  Ans.  18  feet ;  and  15f  ozs. 

74.  Suppose  that  one  of  the  bodies  is  placed  on  a  smooth 
horizontal  table,  and  that  the  string,  by  which  the  bodies  are 
attached,  passes  over  a  smooth  pulley  placed  at  the  edge  of 
the  table  ;  then,  denoting  the  tension  of  the  string  by  T,  we 
have,  as  before, 

Again,  since  the  motion  of  the  body  on  the  smooth  table 
arises  from  the  tension  T,  we  have 

/       T 


62  Action  and  Reaction. 

W 

Eliminating  T,  we  get  /=  ^+  w,g.  (3) 

Again,  equating  the  two  values  of/, 

T=    WW'  .  (4) 

It  may  be  observed  that  the  tension  of  the  string  in  this 
case  is  half  of  that  in  Atwood's  machine  for  the  same  masses. 

75.  Masses  on   Two   Smooth   Inclined   Planes. — 

Suppose  two  bodies,  of  weights  W  and  W\  placed  on  two 
planes,  of  inclinations  i  and  %'  to  the  horizon  ;  and  suppose 
the  connecting  string  to  lie  in  a  vertical  plane  at  right  angles 
to  the  line  of  intersection  of  the  two  inclined  planes,  and  to 
pass  over  a  small  pulley  placed  at  the  common  summit  of 
the  planes ;  then,  representing  as  before  the  tension  of  the 
string  by  T,  since  W  sin  i  is  the  component  of  W  acting 
parallel  to  the  plane,  we  have 

W 
JFrint-jP-— /, 

9 

,     W" 
and  T-W'mii'=—f. 

9 

W  sin  i  -  Wf  sin  x  (KS 

Hence  /=  — — w+  w> 9-  &) 

WW 

Also  T  =  w+  w,  (sin  i  +  sin  i) .  (6) 

It  is  evident  that  W  or  W  will  descend  according  as  W  sin  i 
or  W  sin  i'  has  the  greater  value. 

The  results  of  the  two  former  Articles  are  particular  cases 
of  the  preceding  ;  and  are,  accordingly,  cases  of  the  formulae 
(5)  and  (6) .  We  shall  next  consider  the  preceding  problems 
for  rough  planes. 

76.  Motion  on  Uniformly  Rough  Planes.— Suppose 
two  bodies  connected  as  in  Art.  74,  and  let  fx  denote  the 
coefficient  of  friction  for  the  horizontal  plane. 

The  friction  acting  against  the  motion  of  W  is  represented 


Motion  on  Uniformly  Bough  Planes.  63 

hy  [iW;  hence  the  pressure  producing  motion  is  T-  llW  . 
We  accordingly  have  the  equation 

W 

j 
W 
and  also  W-  T=  — /,  as  before. 

W-  llW 

Hence  we  get  /  =    w+  w,  g,  (7) 

WW 

and  T=W7W'{l+lx)-  (8) 

There  can  be  no  motion  unless  W  is  greater  than  /.t  W ; 
as  is  also  evident  from  elementary  considerations. 
Equation  (7)  may  also  be  written  in  the  form 

from  which  li  can  be  determined  when  W  and  W  are  known, 
/having  been  obtained  by  observation. 

By  this  means  the  value  of  /i,  the  coefficient  of  dynamical 
friction  was  obtained  for  several  substances  by  Coulomb. 

Again,  let  li,  ll  be  the  coefficients  of  dynamical  friction 
for  the  inclined  planes,  in  Art.  75. 

Since  the  pressures  on  the  planes  are  represented  by 
W  cos  i  and  W'  cos  i\  respectively,  the  corresponding  fric- 
tions are  ll  W  cos  i  and  fx  W  cos  i  ;  consequently  the  total 
pressure  acting  on   W,  down  the   plane,  is  represented  by 

W  (sin  i  -  li  cos  i)  -  T ; 

W 

and  we  get  TF(sin  i  -  li  cos  i)  -  T=  — /. 

W 

And,  similarly,    T  -  W  (sin  i  +  li  cos  i')  =  —  /. 

Hence  we  have 

TF(sin  i  -  li  cos  i)  -  W  (sin  %  +  u  cos  i)  .  n  A. 

/= w^w —''  (10) 

WW 

and       T  =  -== — =  { sin  i  +  sin  i  +  li  cos  i  -  li  cos  i) .      (11) 


64  Action  and  Reaction. 


Examples. 

1.  If  the  two  equal  masses  in  Atwood's  machine  be  each  lib.;  required 
the  additional  mass  which,  added  to  one  of  them,  would  generate  a  velocity  of 
one  foot  in  each  mass  at  the  end  of  the  first  second.  ^       2      ^s 

2.  In  the  same  case  find  the  tension  of  the  string  which  connects  the  two 

masses-  Ans.  9-^±  lbs. 

9 

3.  Two  smooth  inclined  planes  are  placed  back  to  back  :  the  inclination  of 
one  is  1  in  7,  and  of  the  other  1  in  10  ;  a  mass  of  20  lbs.  is  placed  on  the  first, 
and  is  connected  by  a  string  with  a  mass  of  30  lbs.  placed  on  the  second  plane. 
Find  the  acceleration  of  the  descent,  and  the  tension  of  the  string. 

4.  A  mass  of  10  lbs.,  falling  vertically,  draws  a  mass  of  15  lbs.  up  a  smooth 
plane,  of  30°  inclination,  by  a  string  pa'ssing  over  a  pulley  at  the  top  of  the 
plane.  Find  the  acceleration,  the  space  fallen  through  in  10  seconds,  and  the 
tension  of  the  string.  ^  /  =  ff_  .  $  =  hg  .   T==  9  lbs> 

5.  A  mass,  descending  vertically,  draws  an  equal  mass  25  feet  in  2£  seconds 
up  a  smooth  plane,  inclined  30°  to  the  horizon,  by  means  of  a  string  passing  over 
a  pulley  at  the  top  of  the  plane.     Determine  the  corresponding  value  of  g. 

*       J  *  Ans.  32. 

6.  Given  the  height,  h,  of  a  smooth  inclined  plane,  find  its  length  so  that  a 
given  weight  F,  descending  vertically,  shall  draw  another  given  weight  Q  up 
the  plane  in  the  least  possible  time.  .  g    2QA 

7.  A  mass  P,  falling  vertically,  draws  another,  Q,  by  a  string  passing  over  a 
fixed  pulley :  if,  at  the  end  of  t  seconds,  the  connecting  string  be  cut,  find  the 
height  to  which  Q  will  ascend  afterwards.  (P-Q\  2  g&_ 

m'    \P+Ql      2  ' 

8.  A  mass,  hanging  vertically,  draws  an  equal  mass  along  a  rough  horizontal 
plane.  If  at  the  end  of  one  second  the  string  be  cut,  find  how  far  the  mass 
will  move  along  the  plane  before  it  is  brought  to  rest  by  the  friction. 

An,  <I^. 
8ft 

9.  In  what  time  will  a  mass  of  2  lbs.,  hanging  vertically,  draw  a  mass  of 
30  lbs.  along  a  smooth  horizontal  table  of  36  feet  length  ?       Ans.  6  seconds. 

10.  If  the  plane  in  the  last  example  be  rough,  and  the  coefficient  of  friction 
be  -h,  rind  the  time  occupied.  Ans.  3  VlO  seconds. 

11.  In  the  previous  example  find  at  what  instant  during  the  motion  the 


Examples.  65 

string  should  be  cut  in  order  that  the  mass  should  just  reach  the  edge  of  the        \, 
table  ;  and  find  the  whole  time  of  motion.  1/ 

AnS'  12  A/T3  seconds>      o  ^  secon<is  {q.p.) 

12.  Two  masses  move  on  two  smooth  inclined  planes,  whose  directions  are  at 
right  angles  to  each  other,  and  are  connected  by  a  string  passing  over  the  inter- 
section of  the  planes.  If  the  tension  of  the  string  be  a  maximum,  find  the  in- 
clination of  either  plane  to  the  horizon.  Ans.  45°.  . 

13.  In  a  single  movable  pulley,  when  there  is  equilibrium,  the  power  and 
the  weight  hang  by  parallel  strings.     The  weight  being  doubled,  and  the  power 
halved,  motion  ensues.     Prove  that,  if  the  friction  and  inertia  of  the  pulley  be        ' 
neglected,  the  tension  of  the  string  will  be  unaltered.     (Catnb.  Trip.,  1874.) 

14.  In  general,  if  Pbe  the  weight  attached  to  the  movable  pulley,  and  Q 
that  to  the  other  end  of  the  string,  prove  that  the  tension  of  the  string  during  the 

motion  is  — — — -  ;  and  that  the  acceleration  of  the  movable  pulley  is  a- 

P+4Q  *       *       F+±Q"' 

the  friction  and  inertia  being  neglected  as  before. 

Let  T  denote  the  tension  of  the  string,  /the  acceleration  of  P,  and/'  that  of 

Q  ;  and  we  have  2P  -  P  =  P  ',     Q  -  T=  Q  "C  ;  but/'  =  2/;  therefore,  &c. 

15.  A  train  is  travelling  at  a  uniform  rate  on  level  rails.  7Fis  the  weight 
of  the  fore  portion  of  the  train,  and  W  that  of  the  brake-van  at  the  end  of 
the  train.  If  the  brakes  be  applied  to  the  brake-van  find  the  stress  produced  on 
the  couplings  between  it  and  the  next  carriage,  assuming  ^  to  represent  the 
coefficient  of  friction.  WW 

Ans.  fx 


W+  W' 

16.  In  Atwood's  machine  if  the  descending  weight  be  a  rigid  homogeneous 
vertical  rod  AB,  prove  that  the  longitudinal  stress  at  any  point  P  of  the  rod 

BP  ' 

during  the  motion,  is  represented  by  —  T,  where  T  is  the  tension  of  the  string. 

17.  In  Atwood's  machine  if  the  pulley  be  rough,  and  if  the  effect  of  friction 
be  to  prevent  motion  until  the  tension  of  the  string  at  one  end  be  greater  than 

that  at  the  other  by  -th  of  the  latter  tension,  prove  that  the  effect  on  the  accele- 
ration will  be  the  same  as  if  the  pulley  remained  smooth  and  the  smaller  weight 
were  increased  by  -  th. 

18.  In  Atwood's  machine,  a  mass  P  is  attached  to  one  end  of  the  string,  and 
two  masses,  Q  and  P,  to  the  other  end,  where  Q  +  P  >  P,  and  P>  Q.  After 
the  united  masses  Q  and  P  have  descended  s  feet  from  rest,  R  is  detached  :  find 
how  much  further  Q  will  move  before  being  brought  to  rest. 

Let/be  the  acceleration  in  the  first  stage  of  the  motion,/'  that  in  the  second, 
v  the  velocity  at  the  instant  P  is  detached,  x  the  required  distance ;  then 


therefore 


2/5  =  p*  = 

2f'x 

; 

/ 

P+  Q 

-  P 

P+  Q 

f 

P+  Q 
F 

+  P' 

P- 

Q 

(     66 


CHAPTER  IV. 

IMPACT    AND    COLLISION. 

77.  Collision  of  Homogeneous  Spheres. — In  this  chap- 
ter it  is  proposed  to  consider  some  elementary  cases  of  impact 
of  solids,  but  principally  the  collision  of  homogeneous  spherical 
bodies,  moving  without  rotation,  whose  centres,  at  the  instant 
of  collision,  move  in  right  lines  lying  in  the  same  plane  (all 
friction  being  neglected). 

There  are  two  cases  to  be  considered,  according  as  the 
centres  of  the  spheres  move  in  the  same  or  in  different  right 
lines.     The  former  is  called  direct,  the  latter  oblique  collision. 

We  commence  with  the  former  case,  and  at  first  suppose 
the  centres  to  move  in  the  same  direction  along  the  line. 

78.  Direct  Collision. — Let  M  and  M  represent  the 
masses  of  the  bodies,  V  and  V  their  velocities  before,  v  and 
v  those  after,  collision.  We  also  suppose  M  to  impinge 
on  M'. 

The  whole  impact  may  be  divided  into  two  stages.  During 
the  first,  the  bodies  compress  each  other,  and  the  impinging 
body  M,  moving  with  a  greater  velocity  than  the  other,  accele- 
rates its  motion,  until  the  exact  instant  at  which  their  mutual 
compression  is  the  greatest,  when  they  are  moving  with 
a  common  velocity.  During  the  second  stage,  the  bodies 
tend  to  revert  to  their  original  shape,  and  the  forces  thus 
brought  into  play,  called  the  forces  of  restitution,  tend  to 
cause  the  bodies  to  separate  by  still  further  diminishing 
the  velocity  of  the  impinging  body,  and  increasing  that  of 
the  other. 

Suppose  u  represents  the  common  velocity  at  the  instant 
of  greatest  compression  ;  then  the  quantity  of  motion  lost  by 
M during  the  first  stage  of  the  shock  is  M(V -  u),  and  that 
gained  by  M'  is  M\u  -  V). 


Collision  of  Homogeneous  Spheres.  67 

These  are  the  measures,  by  Art.  68,  of  the  entire  actions 
of  the  mutual  forces  during  this  stage  of  the  collision  ;  and, 
since  by  the  Third  Law  of  Motion,  the  forces  must  be  equal 
and  opposite,  so  also  are  their  actions  in  the  same  time. 

Hence,  we  have 


H{V-u)  =M'(u-  V), 
MV+M'W 


or  u  = 


M+M' 


(1) 


In  the  case  of  perfectly  non-elastic  bodies,  in  which  no 
forces  of  restitution  are  brought  into  play,  the  bodies  would 
proceed  after  collision  to  move  with  this  common  velocity,     j 

There  is  probably  no  case  in  nature  of  a  perfectly  non- 
elastic  solid.  All  solid  bodies  with  which  we  are  acquainted 
have  a  tendency  to  recover,  in  different  degrees,  their  original 
forms  after  being  compressed.  This  tendency  arises  from 
their  elasticity. 

Bodies  are  said  to  be  perfectly  elastic  when  the  forces  of 
restitution,  brought  into  play  during  the  second  stage,  are 
exactly  equal  to  the  forces  of  compression,  which  act  during 
the  first.  In  this  case  the  impinging  body  M  would  lose 
a  further  quantity  of  motion,  M(V-u),  equal  to  that  which 
it  lost  in  the  first  stage.  Therefore  its  velocity  v,  at  the  end 
of  the  shock,  will  be  equal  to  2u  -  V. 

In  like  manner  we  have  v  =  2u  -  V '.  Thus,  in  direct 
collision,  we  are  enabled  to  determine  the  velocities,  v,  v,  in 
the  case  of  perfectly  elastic  bodies. 

Bodies  are,  however,  in  general  imperfectly  elastic ;  that 
is,  the  whole  force  of  restitution  is  less  than  that  of  compres- 
sion. The  Law  of  restitution,  as  derived  from  experiment,  is 
usually  stated  thus: — The  ratio  which  the  whole  impulse  of 
restitution  bears  to  that  of  compression  is  constant  while  the  imping- 
ing substances  remain  the  same.  This  ratio  is  usually  repre- 
sented by  the  letter  e,  having  been  by  many  writers  called 
the  modulus  or  coefficient  of  elasticity  ;  but  as  this  title  is  now 
employed  in  a  different  sense,  we  shall  follow  the  current 
usage  in  adopting  the  name,  coefficient  of  restitution. 

From  this  law  it  follows  that  the  quantity  of  motion  lost 
by  M  during  the  second  stage  of  the  impact  bears  a  constant 

f2 


68  Impact  and  Collision. 

ratio  to  that  lost  during  the  first ;  and  similarly  for  the  quan- 
tity of  motion  gained  by  M'. 
Accordingly 

M(u-v)=eM{V-u),    M\v'-ii)  =  eM'tu-  V). 

Hence  we  get 

v'-v  =  e(V-V'\  (2) 

and  M  V  +  M'V'=Mv  +  MY.  (3) 

These  equations  enable  us  to  determine  the  velocities, 
v,  v'9  after  impact,  when  those  before  impact  are  given,  as 
also  the  masses  IT,  M',  and  the  coefficient  of  restitution. 

It  should  be  observed  that  equation  (3)  expresses  that  the 
total  quantity  of  motion  of  the  two  bodies  is  the  same  after 
impact  as  before.  This  result  is  a  particular  case  of  a  gene- 
ral principle  which  shall  be  subsequently  considered  (see 
Art.  83). 

The  result  contained  in  (2)  may  be  stated  thus  :  In  direct 
collision*  of  two  spheres  the  relative  velocity  after  collision  bears 
a  constant  ratio  to  the  relative  velocity  before  collision. 

This  law  was  established  by  Newton,  as  the  result  of  ex- 
periment (see  his  Leges  Motus,  scholium) ;  and  the  coefficients 
of  restitution  for  several  substances,  such  as  glass,  ivory, 
steel,  &c,  were  determined  by  him. 

In  more  recent  times  a  number  of  careful  experiments 
were  undertaken  by  Hodgkinson  on  the  laws  of  restitu- 
tion. The  results  are  to  be  found  in  the  Report  of  the  British 
Association,  1834,  and  also  in  the  Transactions  of  the  Royal 
Society.  His  conclusions  agree  in  the  main  with  the  law 
laid  down  by  Newton,  given  above. 

Some  of  the  more  important  results  of  Hodgkinson's 
experiments  may  be  briefly  stated  as  follows  :  — 

The  coefficient  of  restitution  diminishes   slowly   as  the 


*  This  result  is  by  some  writers  taken  as  the  basis  of  the  rational  theory  of 
collision.  However,  the  method  here  given  is  that  more  usually  adopted ;  it  has 
the  great  advantage  of  connecting  the  problem  directly  with  the  consideration  of 
force,  and  of  illustrating  the  principle  (Art.  67)  that  impulsive  forces  must  be 
regarded  as  forces  of  great  intensity  whose  time  of  action  is  very  short. 


Height  of  Rebound.  69 

velocity  of  impact  increases  ;  it  is  independent  of  the  relative 
magnitude  of  the  masses.  In  impact  between  bodies  differ- 
ing much  in  hardness,  the  coefficient  of  restitution  is  nearly 
equal  to  that  between  two  specimens  of  the  softer  body. 

No  perfectly  elastic  body  exists  in  nature  :  glass,  however, 
may  be  regarded  as  nearly  so,  its  coefficient  being  -ff  >  approxi- 
mately, as  determined  by  Newton. 

When  the  mass  Mf  is  at  rest,  and  very  great  in  comparison 
with  M,  v  is  very  small,  and  we  have  approximately  v  =  -  eV. 

Hence,  if  a  body  impinge  perpendicularly,  with  a  velocity 
V,  upon  a  fixed  plane,  it  will  return  back  in  its  former  direc- 
tion of  motion  with  a  velocity  represented  by  e  V,  where  e  is 
the  coefficient  of  restitution. 

79.  Height  of  Rebound. — If  a  body  fall  from  a  height 
h  on  a  fixed  horizontal  plane,  then  V,  the  velocity  with  which 
it  strikes  the  plane,  is  equal  to  y/2gh.  The  velocity  of  rebound 
is  e  V,  or  e  ^2gh :  hence,  if  li  be  the  height  to  which  it  re- 
bounds, we  have 

y2gti  =  e  </2gh, 

or  li  =  e-h.  (4) 

In  all  cases  of  collision  the  student  should  be  careful  to 
give  the  proper  algebraic  signs  to  the  velocities.  The  velocity 
V  of  the  mass  M  is  usually  taken  as  positive  ;  and  hence  the 
other  velocities  will  have  positive  or  negative  signs  accord- 
ing as  they  take  place  in  the  same  or  the  opposite  direction 
to  that  of  V. 

Examples. 

1.  If  a  mass  21  impinge  directly  on  another  mass  21',  at  rest,  find  the  rela- 
tion between  them  when  the  impinging  mass  is  reduced  to  rest  by  the  collision. 

.4ns.  21  —  e2T. 

2.  A  hall  of  6 lbs.  mass,  moving  at  the  rate  of  10  miles  an  hour,  overtakes 
another  of  4  lbs.  mass,  moving  at  5  miles  an  hour :  determine  their  velocities 
after  collision,  assuming  e  =  \;  the  impact  being  supposed  direct. 

Am.  v  =  7;  v'=  9£. 

3.  Find  the  corresponding  velocities  in  the  case  when  the  balls  are  moving 
in  opposite  directions.  Ans.  v  =  1,  v'  =  8^. 


70  Impact  and  Collision. 

~"~  4.  A  mass  of  50  lbs.,  moving  at  the  rate  of  10  feet  per  second,  overtakes 
another  mass  of  25  lbs.,  moving  at  the  rate  of  6  feet  per  second  ;  if  both  masses 
be  perfectly  elastic,  find  their  velocities  after  the  shock. 

Ans.  v  =  7£;  vr  as  ll£. 

5.  A  sphere  impinges  directly  on  a  sphere  of  the  same  mass.  If  they  be 
both  perfectly  elastic,  prove  that  they  interchange  velocities,  after  collision. 

6.  A  mass  drops  from  a  height  of  25  feet  above  a  fixed  horizontal  plane,  and 
rebounds  to  a  height  of  9  feet ;  find  the  coefficient  of  restitution.  e  =  f . 

7.  A  mass  of  10  lbs.,  moving  with  a  velocity  of  10  feet  per  second,  impinges 
directly  on  another  of  5  lbs.,  supposed  at  rest.  If  the  coefficient  of  restitution 
bef,  and  the  duration  of  collision  be  rtyoth  part  of  a  second,  determine  the  mean 
value  of  the  mutual  pressure  between  the  balls  during  the  collision. 

Here,  we  easily  find  u'  =  12.  Hence,  by  Art.  41,  we  find  that  the  pressure 
which,  acting  for  Tootn  or"  a  second,  would  generate  a  velocity  of  12  feet  per 
second,  in  a  body  of  5  lbs.  mass,  is  187^  lbs. 

8.  An  imperfectly  elastic  sphere  falls  from  a  given  altitude  above  a  hori- 
zontal plane,  and  rebounds  continually:  find  the  whole  space  described;  and 
also  the  whole  time  before  it  is  brought  to  rest ;  neglecting  the  time  occupied  by 
the  series  of  impacts,  and  also  the  resistance  of  the  air. 

Let  a  denote  the  given  altitude,  and  s  the  whole  space  described. 
Then  the  height  of  the  first  rebound  is  ae1;  that  of  the  second  aei)  &c. 

1  +e2 
Therefore  *  =  a  +  2ae2  +  2aei  +  &c.  =  a  2. 

Again,  if  V,  Vi,  F2,  &c,  F„,  be  the  velocities  with  which  the  sphere  strikes 
the  plane,  at  the  different  impacts,  we  have  j 

Vi  =  eV,     V2  =  eVi  =  e'iV,     &c,  Vn=  enV. 

Also  let  t,  th  hi  &c.,  tn,  be  the  corresponding  intervals  of  time  ;  then 

V  2Fi      27  2V  . 

t=-,     ti  = =  — e,    h  = — <?2,&c. 

9  9         9  9 

Hence,  the  entire  time  T  is  given  by  the  equation 

v  „     „   /    n  +  « 

T=  —  ll  +2e+  2ei  +  &c.)  =  —  - • 

9  '       g  \-e 

80.  Oblique  Collision.— We  now  proceed  to  the  case 
of  oblique  collision,  i.  e.  where  the  centres  of  the  spheres  are 
not  moving  in  the  same  right  line. 

We  shall  suppose  the  spheres  to  be  homogeneous,  and 
perfectly  smooth  ;  so  that  their  entire  mutual  action  and  re- 
action has  place  along  the  common  normal  at  their  point  of 
contact,  that  is  the  line  connecting  the  centres  of  the  spheres. 
We  also  suppose  that  the  lines  along  which  the  centres  of  the 
spheres  are  moving  before  collision  lie  in  the  same  plane. 


Oblique  Collision.  71 

Let  V,  V  be  the  velocities  at  the  instant  of  impact ;  and 
«,  a  the  angles  which  the  line  joining  the  centres,  at  the 
commencement  of  the  collision,  makes  with  the  respective 
directions  of  motion. 

Let  v,  v;  j3,  j3'  be  the  corresponding  velocities  and  angles 
at  the  end  of  the  collision. 

Eesolve  V  into  its  components,  V  cos  a  and  V  sin  a,  re- 
spectively along  and  perpendicular  to  the  right  line  join- 
ing the  centres.  Make  a  similar  resolution  of  the  velocities 
after  collision.  Then,  since  the  forces  brought  into  play 
during  the  collision  act  along  the  line  joining  the  centres, 
the  velocities  perpendicular  to  that  line  are  unaffected  by  the 
collision. 

Hence  we  have 

V  sin  a  =  v  sin  /3,      V  sin  a  =  v  sin  j3'.  (5) 

Again,  the  component  velocities  Fcosa,  &c,  along  the 
line  joining  the  centres  of  the  spheres,  will,  by  the  Laws  of 
Motion,  be  subject  to  the  same  relations  (2),  (3),  as  those 
already  established  for  direct  collision — hence,  we  obtain  the 
two  additional  equations 

tf  cos  j3'  -  V  cos  |3  =  €  (  V  COS  a  -  V  COS  a).  (6) 

M Fcos  a  +  M '  V  cos  a  =  Mv  cos  j3  +  M'v  cos  j3'.     (7) 

These,  along  with  the  two  preceding  equations  (5),  are 
sufficient  for  the  determination  of  the  velocities  and  the  direc- 
tions of  motion  after  impact,  when  the  corresponding  velo- 
cities and  directions  before  impact  are  known,  as  also  the 
masses  and  the  coefficient  of  restitution. 

In  the  case  of  oblique  collision  of  a  sphere  against  a 
smooth  fixed  plane,  let  Vhe  the  velocity  of  the  sphere  before 
collision,  and  a  the  angle  its  direction  of  motion  makes  with 
the  perpendicular  to  the  smooth  plane ;  then  Fcos  a  repre- 
sents the  velocity  perpendicular,  and  V  sin  a  that  parallel,  to 
the  fixed  plane. 

If  v  and  /3  represent  the  corresponding  velocity  and 
angle  after  collision,  since  the  velocity  parallel  to  the  smooth 
plane  is  unaltered  by  collision,  we  have 

t'sin/3  =  Fsina.  (8) 


72  Impact  and  Collision. 

Again,  the  velocity  perpendicular  to  the  plane  will  be  affected 
in  the  same  manner  as  in  direct  collision,  and  we  accordingly 
have 

flcos/3  =  eVcosa,  (9) 

Hence,  by  division,  we  get 

tana  =  etanj3,  (10) 

which  gives  the  direction  of  motion  after  impact. 

The  angles  a,  (3  are  sometimes  called  the  angles  of  inci- 
dence and  reflexion ;  and  the  preceding  result  shows  that  the 
tangents  of  these  angles  are  to  each  other  in  the  constant  ratio 
of  the  coefficient  of  restitution  to  unity.  These  angles  are 
equal  in  the  case  of  perfectly  elastic  bodies. 

The  subsequent  motion  of  the  body  depends  on  the 
continuous    forces    which    act    on     it.  n 

When  gravity  is  the  only  acting  force 
the  path  is  a  parabola,  as  in  Art.  48 ; 
and  the  parabolic  path  is  determined 
from  the  initial  velocity  v,  and  the 
initial  direction  of  motion,  /3. 

Examples. 

1 .  If  the  mass  31'  be  at  rest  before  collision,  find  the  directions  of  motion 
after  collision. 

,     n  31+ M' 

Am.  /3  =  0  ;   tan  ;3  =  — —  tan  a. 

M-eM' 

2.  A  perfectly  elastic  ball  impinges  obliquely  on  another  of  equal  mass  at 
rest,  prove  that  the  directions  of  their  motions  after  impact  are  at  right  angles 
to  one  another. 

3.  A  ball  impinges  on  another  at  rest ;  prove  that  if  the  coefficient  of  resti- 
tution be  equal  to  the  ratio  of  their  masses  the  balls  will  move  in  directions  at 
right  angles  to  each  other,  whatever  be  the  direction  of  the  impact. 

4.  How  is  this  statement  to  be  modified  in  the  case  of  direct  collision  ? 
The  impinging  ball  is  brought  to  rest  by  the  collision. 

5.  A  ball  is  reflected  in  succession  by  two  fixed  smooth  planes  of  the  same 
substance,  which  are  at  right  angles  to  one  another ;  the  ball  moves  in  a  plane 
at  right  angles  to  the  intersection  of  the  fixed  planes.  Prove  that  the  direction 
of  motion  before  the  first  and  after  the  second  reflection  are  parallel. 

6.  A  projectile  strikes  a  perfectly  elastic  wall,  which  is  perpendicular  both 
to  the  horizon  and  to  the  plane  of  the  projectile's  flight :  find  the  horizontal 
range  of  the  reflected  projectile. 

Since  the  angles  of  incidence  and  reflection  are  equal  in  this  case,  as  also 
the  velocities  before  and  after  impact,  it  is  evident  that  the  parabolic  path  of  the 


Vis  Viva  of  a  System.  73 

projectile  after  striking  the  wall  is  equal  in  every  respect  to  that  which  it  would 
have  continued  to  describe  if  there  had  been  no  wall  interposed.  Accordingly 
the  problem  is  solved  by  aid  of  Art.  50. 

7.  An  imperfectly  elastic  particle  is  projected  from  a  point  in  a  smooth  hori- 
zontal plane"  with  a  given  velocity  V,  and  in  a  given  direction  a,  and  proceeds 
to  describe  a  series  of  parabolic  paths  by  a  number  of  rebounds  from  the  plane  : 
find  the  whole  time  elapsed  before  it  ceases  to  rebound  ;  and  also  its  subsequent 
motion. 

Resolve  the  velocity  of  projection  into  vertical  and  horizontal  components, 
Fsin  a  and  V  cos  a. 

The  horizontal  component  Fcos  a  will  be  unaltered  by  the  successive  impacts, 
and  accordingly  remains  constant  throughout  the  motion :  the  vertical  component 
Fsin  o  will  be  altered  at  each  impact  in  the  same  manner  as  in  direct  collision  ; 
accordingly  it  may  be  treated  as  in  Ex.  8,  p.  70. 

Hence,  if  The  the  entire  time  before  the  vertical  velocity  Fsin  a  is  destroyed, 
we  easily  get,  as  in  the  example  referred  to, 

2  V  sin  a  J_ 

g       l-e 

Again,  since  the  horizontal  velocity  is  constant,  and  equal  to  Fcos  a,  the  whole 
range  before  the  vertical  velocity  ceases  is 

V2  sin  la 


<7(1  "  e) 
The  body  would  subsequently  move  along  the  plane  with  the  constant  velocity 
Fcos  a. 

It  should  be  observed  that  the  particle,  in  this  problem,  describes  a  series  of 
parabolic  curves,  one  for  each  rebound. 

81.  Yis  Viva  of  a  System. — If  each  point  of  a  mass  m 
be  moving  with  the  same  velocity,  and  if  v  denote,  at  any 
instant,  the  velocity  common  to  all  its  points,  then  the  quantity 
represented  by  mv2  is  called  the  vis  viva  of  the  mass  at  the 
instant.  In  general,  in  the  motion  of  any  system  of  masses, 
if  each  element  of  mass  be  multiplied  by  the  square  of  its 
velocity  at  any  instant,  and  the  sum  of  these  products  taken 
for  the  entire  system,  this  sum  is  called  the  vis  viva  of  the 
system  at  that  instant.     It  is  represented  by  the  expression 

*2(mv2). 

It  is  easily  seen  that,  in  perfectly  elastic  Sjiheres,  the  Vis 
Viva  is  unaltered  by  Collision. 

For,  since  e  -  1  in  this  case,  equation  (2)  becomes 
V+v=  V'+  v;  also,  we  have 

M(V-v)  =M'(v'-V). 


74  Impact  and  Collision. 

Multiplying,  we  get 

M{V2-v2)  =  M'{v2-  F'2), 
or  MV2  +  M'  V'2  =  Mv2  +  M'v2.  (11) 

Hence,  in  direct  collision  between  elastic  spheres  the  vis 
viva  is  the  same  after  collision  as  before. 

It  can  be  easily  seen  that  the  same  principle  holds  in  the 
oblique  collision  of  perfectly  elastic  spheres.  For  the  preced- 
ing demonstration  holds  for  the  components  of  velocity  esti- 
mated along  the  line  j  oining  the  centres  of  the  spheres  at  the 
instant  of  collision :  moreover,  the  tangential  components  of 
velocity  are  unaffected  by  collision ;  consequently  (since 
V2  =  V2  sin2a  +  V2  cos*a,  &c.)  it  follows,  in  the  case  of  perfect 
elasticity,  that  the  vis  viva  is  unaltered  by  collision. 

82.  Momentum  of  any  System. — Let  (x9  y9  z),  [x ,  y'9  z)9 

(x"9  y"9  s"),  &c,  at  any  instant,  denote  the  coordinates  of  a 
number  of  moving  particles,  m,  m',  m",  &c,  referred  to  a  fixed 
system  of  rectangular  axes  ;  then,  by  Art.  12,  the  component 
velocities  of  m,  at  the  instant,  parallel  to  the  axes  of  x9  y9  z, 
are  x,  y,  z,  respectively. 

Again,  resolving  the  quantity  of  motion  of  m,  in  the  same 
directions,  we  get  for  its  components  the  expressions 

dx        dy        dz 
mx,my,mn,  ov  m -,  m-,  mJf 

If  this  be  done  for  the  other  masses  m9  m",  &c,  the  whole 
quantity  of  motion  or  momentum  of  the  system,  at  the  instant, 
estimated  parallel  to  the  axis  of  x,  is  represented  by  2»u. 

In  like  manner,  the  whole  momentum  parallel  to  the  axes 
of  y  and  z  are  ^my  and  Smz,  respectively. 

Again,  let  x,  y,  z  represent  the  coordinates  of  the  common 
centre  of  inertia  of  the  system  m9  m9  &c,  at  the  same  in- 
stant, we  have,  denoting  the  sum  of  the  masses  by  M9 

Mx  =  *2mx,   My  =  *2my9   Mz  =  Sms. 

Moreover,  since  these  equations  are  true  throughout  the 


Conservation  of  Momentum.  75 

motion,  we  may  differentiate  them,  with  respect  to  the  time 
f,  and  thus  we  obtain  the  equations 

.-dx      „     dx     _    . 
M-T-  =  Sw  —  =  2  ww 

Hence  the  resolved  part  of  the  momentum  of  a  system  in 
any  direction  is  equal  to  the  whole  mass  of  the  system  mul- 
tiplied into  the  component  of  the  velocity  of  the  centre  of 
gravity,  in  the  same  direction. 

83.  Conservation  of  Momentum. — It  is  easily  seen 
that  the  momentum  in  any  direction  of  any  system  of  bodies 
is  unaltered  by  their  mutual  collision.  For,  under  all  cir- 
cumstances of  collision,  the  actions  and  reactions  are  equal 
and  opposite  ;  and  as  these  forces  are  measured  by  the  quan- 
tities of  motion  which  they  generate  or  destroy,  it  follows, 
whenever  two  bodies  of  the  system  come  into  collision,  that 
whatever  momentum,  in  any  direction,  is  generated  by  the 
impact  in  one  of  the  impinging  bodies,  an  equal  momentum 
in  the  same  direction  must  be  destroyed  in  the  other.  So  that 
the  entire  momentum,  in  that  direction,  is  unaltered  by  the 
collision.  The  same  holds  whatever  number  of  collisions  be 
supposed  to  take  place  between  the  members  of  the  system. 

Hence,  we  infer  that  the  entire  momentum  of  the  system, 
resolved  in  any  direction,  is  unaffected  by  impacts  among  the 
parts  of  the  system. 

It  can  be  seen,  without  difficulty,  that  the  same  mode 
of  reasoning  applies  to  any  case  of  internal  mutual  action 
between  the  several  parts  of  the  system,  whether  arising 
from  attractions,  molecular  forces,  or  otherwise  :  since,  in  all 
cases,  to  every  action  corresponds  an  equal  and  opposite  re- 
action. 

"We  accordingly  infer  that,  if  a  system  be  subjected  only  to 
the  internal  mutual  forces  between  the  bodies  which  constitute  it, 


76  Impact  and  Collision. 

the  total  resolved  momentum  in  any  direction  is  constant;  i.e. 
Imx,  *2my,  2*W2,  have  constant  values  during  the  motion. 

84.  Conservation*  of  Motion  of  Centre  of  Inertia. 

— It  follows  from  (12)  that  -^,  -j-,  — ,  i.  e.  the  component 

at    at    at 

velocities  of  the  centre  of  inertia  of  a  system,  will  be  constant 

throughout  the  motion  whenever  the  quantities  of  motion 

2w  x ,  1m  y ,  S?«  s ,  are  constant. 

Hence,  from  the  preceding  Article,  it  follows  that  the 
velocity,  and  also  the  direction  of  motion  of  the  centre  of 
inertia  of  any  system,  are  constant,  whenever  the  system  is 
subject  only  to  the  mutual  actions  and  reactions  of  the  bodies 
which  constitute  it. 

This  is  a  generalization  of  the  principle  of  inertia  con- 
tained in  the  First  Law  of  Motion,  and  may  be  otherwise 
stated  thus  : — A  system  of  bodies  cannot  by  their  mutual  actions 
and  reactions  alter  the  motion  of  their  common  centre  of  inertia. 

Hence,  in  such  a  system,  when  not  acted  on  by  any 
external  forces,  the  common  centre  of  inertia  must  either 
remain  at  rest  or  move  uniformly  in  a  right  line. 

85.  The  general  principle,  that  the  entire  quantity  of 
motion  of  two  or  more  bodies  is  unaltered  by  their  mutual 
actions  and  reactions,  furnishes  us  with  a  ready  method  of 
solving  some  elementary  problems. 

For  example,  suppose  two  masses,  m  and  m',  to  be  con- 
nected by  a  string,  and  laid  on  a  perfectly  smooth  horizontal 
table,  at  a  distance  from  each  other  less  than  the  length  of 
the  string.  Now,  let  a  given  impulse  be  applied  to  m  along 
the  line  which  joins  it  to  m\  the  motion  which  ensues  after 
the  string  becomes  stretched  can  be  easily  found  as  follows : 

Let  m  V  be  the  quantity  of  motion  communicated  to  m 
by  the  impulse,  then,  after  the  string  becomes  tight,  the 
bodies  must  move  with  a  common  velocity.  Let  Vi  denote 
this  common  velocity ;  then,  since  the  whole  quantity  of 
motion  of  the  two  bodies  remains  the  same,  we  have 

(m  +  m)  i\  =  m V. 


*  This  proof  corresponds  in  the  main  with  that  given  hy  Newton. 
Leges  Motus,  Cor.  iv. 


Conservation  of  Motion  of  Centre  of  Inertia.  77 

Consequently  they  move  along  the  line  with  a  common 
velocity 

mV 

m  +  m' 

In  this  problem  we  have  supposed  the  motion  one  of  pure 
sliding ;  and  we  neglect  the  mass  of  the  string  in  it  as  also 
in  the  next  problem. 

86.  A  mass  IT,  after  falling  through  a  height  h,fro?n  the 
edge  of  a  smooth  table,  commences  to  draw  by  an  inextensible 
string  another  mass  M\  which  rests  on  the  table;  to  find  the 
velocity  communicated  to  M'  at  the  instant  that  the  string  be- 
comes  tightened,  and  also  the  impulse  of  the  tension  of  the  string. 

The  velocity  acquired  by  M,  in  consequence  of  its  fall,  is 
represented  by  y/2gh ;  and  since  at  the  end  of  the  impulsive 
strain  the  bodies  are  moving  with  equal  velocities,  and  also 
the  quantity  of  motion  is  unaltered  by  the  impulsive  action, 
we  must  have 

(Jf  +  M')t\  =  MV=  M*/fyh9 

or  i\  =  M+M,  </fyh>  (13) 

where  vx  denotes  the  common  velocity  at  the  instant  in  ques- 
tion. 

Again,  the  impulse  of  the  tension  of  the  string  is  measured 
by  the  quantity  of  motion  communicated  to  M'\  and  accord- 
ingly is  represented  by 

If  the  table  be  rough,  since  the  friction  of  the  table  is  pro- 
portional to  the  weight  of  IT,  it  may  be  neglected  in  com- 
parison with  the  impulsive  force,  and  we  obtain  the  same 
value  for  i\  as  in  the  case  of  a  smooth  table  (see  Art.  67). 


J 


78  Impact  and  Collision. 


Examples. 

1.  A  sphere,  of  30  lbs.,  moving  with  a  velocity  of  45  feet  a  second,  over- 
takes another,  of  27  lbs.,  moving" 32  feet  a  second  ;  if  the  relative  coefficient 
of  restitution  be  f,  find  their  velocities  after  collision.  Ans.  34if,  43ff . 

2.  Two  spheres  meet  directly  with  equal  velocities  ;  find  the  ratio  of  their 
masses  that  one  of  them,  M,  should  be  reduced  to  rest  by  the  collision— (1)  when 
perfectly  elastic ;  (2)  for  coefficient  of  restitution  e. 

Am.  (1)  M  =  3.1/',  (2)  M  =  31'  (1  +  2e). 

3.  If  two  equal  and  perfectly  elastic  spheres  be  dropped  at  the  same  instant 
from  different  heights,  h  and  h',  above  a  horizontal  plane ;  determine  whether 
their  common  centre  of  inertia  will  ever  rise  to  its  original  height. 

Ans.  No,  unless  A—  is  a  commensurable  number. 

4.  A  101b.  shot  is  fired  from  a  gun  of  12cwt,  that  is  quite  free  to  move. 
The  velocity  with  which  the  shot  leaves  the  mouth  of  the  gun  is  1600  feet  per 
second  ;  find  the  velocity  of  the  gun's  recoil.  Ans.  11*9  feet  per  second. 

5.  Three  homogeneous  spherical  bodies,  m,  m',  m",  are  placed  with  their 
centres  in  a  row.  If  m  be  projected  with  a  given  velocity  V  towards  m;  to  find 
the  magnitude  of  m'  in  order  that  the  velocity  communicated  to  m"  by  its  inter- 
vention shall  be  the  greatest  possible. 

Let  e,  e  denote  the  relative  coefficients  of  restitution  between  m,  m,  and 
hetween  m',  m",  respectively.  Then,  if  v'  be  the  velocity  of  m'  after  the  first 
collision,  we  get  from  Art.  78, 

,      m{l  +  e)V 
v  =      '—. 

m  +  m 

In  like  manner,  if  v"  be  the  velocity  of  m"  after  the  second  collision,  we  have 

„ _  tn'(l  +  e')v'  _  mm  (I  +  e)  (1  +  e)  V 
m'  +  m"     ""   (m  +  m)  (w'  +  m") 

m'  ,  .  (*«  +  ♦»')  {m'  +  m") 

Accordingly,  ; 7V1— jjz  must  be  a  maximum  ;  or —, 

°        {m  +  m')  {m'  +  m  )  m 

mm"  ...  ,     mm"  •  •  • 

is  a  minimum :  i.e.  m  +  m  +  m"  +  — —  is  a  minimum,  or  m  +  — -  is  a  mini- 

tn  m 

mum:  hence,  m  =  Vw»»",  by  elementary  algebra  ;  consequently  the  masses  must 
be  in  geometrical  progression. 

This  reasoning  is  readily  extended  to  the  case  of  any  number  of  spheres 
placed  in  a  row  ;  and,  when  the  first  and  last  are  given,  the  masses  must  be  in 
geometrical  progression,  in  order  that  the  velocity  communicated  to  the  last 
should  be  the  greatest  possible. 


Examples.  79" 

6.  Two  particles  are  connected  by  a  string,  and  laid  on  a  uniformly  rough 
horizontal  table,  at  a  distance  from  each  other  less  than  the  length  of  the  string. 
One  of  the  particles  receives  a  given  impulse  along  the  line  joining  them  :  deter- 
mine the  motion  which  ensues  after  the  tightening  of  the  string. 

7.  Find  an  expression  for  the  vis  viva  lost  in  the  direct  collision  of  two 
imperfectly  elastic  spheres. 

From  equations  (2),  (3),  Art.  78,  we  have 

(mV  +  m'Y')2  =  (mv  +  m'v')2, 
and  mm'e~(V  —  F')2=  mm'(v  -  v')2. 

This  latter  may  be  written 

mm\V  -  r')2=mm'(v-v')2  +  (1  -  f)mm'(V-  V'f. 
Hence,  by  addition, 

(m  +  m')(mV2  +  m'V2)  =  (m  +  m')(mv2  +  m'v'2)  +  (1  -  e2)mm'(V-  V'f . 

Therefore      m  V2  +  *»'  V'2  =  mv2  +  m'v'2  +  (1  - e2) ;  ( V-  F')2. 

m  +  m 

Accordingly,  the  vis  viva  lost  by  the  collision  is  represented  by 

(1_,)_=£_(r_F7. 

v  Jm  +  m' v  ; 

8.  Find  the  loss  of  vis  viva  caused  by  the  direct  impact  of  two  balls,  one 
weighing  10  lbs.  and  falling  from  a  height  of  20  feet,  the  other  at  rest  and 
weighing  301b.  ;  assuming  the  coefficient  of  restitution  =  -|. 

Ans.  ^-th  of  the  original  vis  viva. 

9.  A  body,  after  sliding  down  a  smooth  inclined  plane  of  given  height,  re- 
bounds from  a  hard  horizontal  plane ;  find  the  range  on  the  latter  plane. 

10.  A  mass  31,  after  falling  freely  through  h  feet,  begins  to  pull  up  a 
heavier  mass  Mi  by  means  of  a  string  passing  over  a  pulley,  as  in  At  wood's 
machine  ;  find  the  height  through  which  it  will  lift  it. 

Let  v\  be  the  velocity  communicated  to  M\  by  the  impulsive  action  ;  then  by 

Art.  86  we  have  v\  =  — —  ^l2gh. 

M  -f  M\ 

During  the  subsequent  motion  Mi  is  subject  to  a  uniform  retardation  — — —g, 

as  in  Art.  73;  accordingly,  if  ITdenote  the  height  to  which  iLTi  ascends  before  it  is 
brought  to  rest,  we  have 

2/     Mr-M- 

11.  An  inelastic  particle  falls  from  rest  to  a  fixed  inclined  plane,  and  slides 
down  the  plane  to  a  fixed  point  in  it ;  show  that  the  locus  of  the  starting  point 
is  a  straight  line  when  the  time  to  the  fixed  point  is  constant.  (Camb.  Trip., 
1871). 


SO  Impact  and  Collision. 

12.  Two  equal  balls  of  radius  a  are  in  contact  and  are  struck  simultaneously 
by  a  ball  of  radius  c  moving  in  the  direction  of  their  common  tangent ;  if  all  the 
balls  be  of  the  same  material,  the  coefficient  of  elasticity  being  e,  find  the  velo- 
cities of  the  balls  after  impact,  and  prove  that  the  impinging  ball  will  be  reduced 
to  rest  if 

2e=;"+  C\.  (Camb.  Trip.,  1871.) 

ad(2a+c)  ' 

13.  Show  how  to  determine  the  motion  of  two  elastic  spheres  after  direct 
\S        impact,  and  prove  that  the  relative  velocity  of  each  of  them  with  regard  to  the 

•centre  of  mass  of  the  two  is,  after  the  impact,  reversed  in  direction  and  reduced 
in  the  ratio  e :  1,  e  being  the  coefficient  of  restitution. 

A  series  of  n  elastic  spheres  whose  masses  are  1,  e,  e2,  &c,  are  at  rest,  sepa- 
rated by  intervals,  with  their  centres  on  a  straight  line.  The  first  is  made  to 
impinge  directly  on  the  second  with  velocity  u.  Prove  that  the  final  vis  viva  of 
the  system  is  (1  -  e  +  en)u2.  {Ibid.,  1875.) 

14.  An  elastic  ball  makes  a  series  of  rebounds  from  a  perfectly  smooth  inclined 
plane  :  to  investigate  its  motion. 

Let  i  be  the  inclination  of  the 
plane  to  the  horizon,  and  suppose 
the  ball  projected  from  the  point  0 
in  the  plane,  in  a  direction  which 
makes  the  angle  a  with  the  plane. 
Let  £,  jBi,  &c,  fin  be  the  angles  at 
which  the  ball  strikes  the  plane  at 
the  first,  second,  .  .  .  nth  impacts ;  and 
oi,  ct2,  •  •  •  a.,,,  the  angles  it  makes 
after  rebounding. 

Then,  by  equation  9,  Art.  56,  we  have 

cot  £  =  cot  o  -  2  tan  i ; 

cot  £  =  e  cotoi, 
.-.     ecotai  =  cot  a  —  2tani. 
Similarly  e  cot  02  =  cot  01  —  2  tan  i ; 

.-.     tf2coto2  =  cot  a  -  2(1  +  «)tani; 
and  it  is  easily  seen  that  we  have,  in  general, 

en  cot  «„  =  cot  o  -  2  (1  +  e  +  .  .  .  +  e'1'1)  tan  i 

2(1  -*»).      • 

=  cot  o \ tan  t, 

1  -  e 

from  which  the  angle  after  the  nth  rebound  can  be  found. 

Again,  the  ball  will  proceed  to  bound  up  the  plane  so  long  as  the  angles 
ai,  «*■>,..  .  are  each  less  than  90°  —  i. 


but,  by  (10)  Art.  80, 


Examples.  81 

If  a»  be  the  first  of  a  series  of  angles  which  exceeds  90°  -  i,  we  will  have 
cot  o»<  tan  t. 

If  cot  a  is  greater  than it  can  be  readily  shown  that  for  all  values  of  n 

an  is  less  than  90°  -  i ;  and  in  this  case  accordingly  the  ball  would  proceed  to 
ascend  the  plane  by  au  indefinite  series  of  parabolic  paths. 

But  if  cot  a  be  less  than  — — ,  after  a  certain  number  of  impacts,  the  body 

would  proceed  to  rebound  down  the  inclined  plane. 
T      .  .     .  .  2  tan  i 

In  the  particular  case  where  cot  a  = ,  or  2  tan  i  =  cot  a  ( 1  —  e),  we  have 

1  —  e  v         " 

e  cot  oi  =  e  cot  o  ;     .'.01  =  0; 

hence  a  =  a\  =  az  =  . .  .  =  an  ; 

or,  all  the  angles  of  rebound  are  equal  to  one  another  ;  consequently  the  series 
of  parabolic  paths  in  this  case  are  similar,  and  the  particle  would  proceed  up  the 
plane  with  an  indefinite  number  of  rebounds. 

In  general,  let  t\t  fa,  .  .  .  t„  be  the  times  of  flight  for  the  series  of  parabolic 
paths,  and  v\,  02,  .  •  •  vn,  the  velocities  of  the  successive  rebounds ;  then  by  equa- 
tion (5),  Art.  50,  we  have 

2v  sin  a  2v\  sin  a\    B 

t\  =  r,     h  = — ,  &c. 

g  cos  1  g  cos  1 

But  if  v'  be  the  velocity  with  which  the  ball  strikes  the  plane  at  the  first 
impact,  we  have 

v'  sin  /3  =  v  sin  a ; 
but  by  Art.  80, 

vi  sin  a\  -  ev'  sin  $  =  ev  sin  a ; 
consequently 

h  =  et\ ;  also  h  =  eh  =  e2ti,  &c. 

Hence  the  times  of  flight  are  in  geometrical  progression,  having  e  for  their 
common  ratio. 

If  the  intervals  of  time  occupied  by  the  successive  impacts  be  neglected,  we 
get  for  the  time  T  of  describing  the  first  n  parabolas, 

2v  sin  o   1  -  en 
g  cos  i     1  -  e  ' 

Again,  let  i?i,  i?2,  .  .  .  B,i  denote  the  consecutive  ranges" on  the  inclined 
plane  ;  then,  by  Art.  50,  we  have 

-Si  =  ?gh2  — : =  \gtf-  cos  i  (cot  a  -  tan  i). 

sin  a  ' 

Similarly, 

Hz  =  \gh*  cos  i  (cot  a\  -  tan  i)  =  \gt£  cos  i  {e2  cot  ai  -  e2  tan  i) 

=  yeh*  cos  i  {  cot  a  -  (2  +  e)  tan  i } . 
G 


82  Impact  and  Collision. 

And,  in  general, 

Rn  =  lgtn2  cos  i  (cot  ctn-i  —  tan  i)  ^ 

=  ye»-1ti2cosi{cota-(2  +  2e+  . .  .  +  2e«-2  +  e"-1)  tani} 

(2  —  £n-1  —  e"  \ 
1 ) 

2v2sin2a  (      ,  2e"~\       .      1  +  *  .   _.       .) 

= \en~l  cot  a tant  +  - e2»-2tantj. 

y  cos i     (  1  —  e  1  -  e  ) 

Hence  the  sum  of  w  ranges  is  found  to  be 

n  v2  sin2a  1  -««  (  1  -e»        .) 

=  2 r  -r j  cot  a  - tan  x  \ 

g  cos  t    1  -  e   (  1  -  e  ) 

(  1  -  en  A 

=  t>T  sin  a  (  cot  a  — ■ tan  i  ) . 

\  1  -e  J 

2  tan  i 
If  cot  a  be  greater  than we  get  the  entire  range  on  the  inclined  plane 

by  making  n  —  oo  ;  hence  the  entire  range  is,  in  this  case, 

rr    •         (      *           tani 
v  1  sin  a  \  cot  o 

I  l-« 

The  preceding  question  was  discussed  at  great  length  by  Bordoni,  Mem.  della 
Societa  Ital.,  1816.  See  also  Walton's  Problems  on  Theoretical  Mechanics, 
pp.  262,  263,  3rd  edition. 

15.  In  the  preceding  example  show  that  the  greatest  distances  of  the  body 
from  the  inclined  plane  in  the  successive  parabolic  paths  are  in  geometrical  pro- 
gression, having  e1  as  their  common  ratio. 

16.  If  two  bodies,  of  the  same  elasticity  be  projected  with  the  same  velocity 
from  a  point  on  an  inclined  plane,  and  if  the  directions  of  projection  make  equal 
angles  at  opposite  sides  of  the  perpendicular  to  the  plane,  prove  that  the  series 
of  parabolic  paths  described,  one  up,  the  other  down  the  plane,  will  be  described 
in  times  which  are  respectively  equal  in  pairs. 

17.  An  imperfectly  elastic  ball  falls  from  a  height  h  upon  an  inclined  plane ; 
find  the  range  between  the  first  and  second  rebounds.     Am.  4eh  sin  i  (1  +e). 

18.  Prove  that,  in  order  to  produce  the  greatest  deviation  in  the  direction  of 

a  smooth  billiard  ball  of  diameter  a,  by  impact  on  another  equal  ball  at  rest,  the 

a    ll-e 

former  must  be  projected  in  a  direction  making  an  angle  sin-1  -   / with  the 

c  \  3  —  e 

line  (of  length  c)  joining  the  two  centres ;  e  being  the  coefficient  of  restitution. 

Camb.  Trip.,  1873. 

19.  A  bucket  and  a  counterpoise,  connected  by  a  string  passing  over  a  pulley, 
just  balance  one  another,  and  an  elastic  ball  is  dropped  into  the  centre  of  the 
bucket  from  a  distance  h  above  it ;  find  the  time  that  elapses  before  the  ball 


Examples.  83 

ceases  to  rebound ;  and  prove  that  the  whole  descent  of  the  bucket  during  this 
imh  e 

interval  is  — where  m,  M  are  the  masses  of  the  ball  and  the 

2ji  +  m   (1  —  ey 

bucket,  and  e  is  the  coefficient  of  restitution.  Camb.  Trip.,  1875. 

Let  v  be  the  velocity  of  the  ball  just  before  the  first  impact.  The  relative 
velocity  after  the  first  impact  is  ev,  and  the  relative  acceleration  is  g,  since  the 
acceleration  of  the  bucket  is  zero. 

Therefore  the  time  during  which  the  ball  rebounds  is 


2»  #        ,  K      2v      e 

—  (e  +  e2  +  ez  +  . .  . )  = 

9  U   1  -  e        1  -e\  g 


e       hh 


Let  Pi,  r2,  r3,  ...  be  the  velocities  of  the  bucket  between  the  first,  second, 
third,  .  .  .  impacts. 

„,  „      m(l+e)  m(l+e) 

Then  7  i  =     v  y>     Fa  =  Pi  +  -± '-  ev,  &c. , 

2M  +  m  2JI  +  m 

and  the  space  described  by  the  bucket  is 

2v  .  _        _  Tr        ,  TT  ,  2mev2  4mh  e 

—  (eVi  +  e2  V*+  63  r3  +  . .  •)  = 


g  v  '      ^(2Jf+«i)(l-«)2      21T+m   (1  -  e) 


(This  proof  is  taken  from  GreenhilTs  solutions  of  Cambridge  Problems   and 
Eiders  for  1875.) 

20.  A  particle  is  projected  with  a  velocity  V,  in  a  direction  making  an  angle 
a  with  the  horizon,  and  strikes  a  vertical  wall,  at  a  distance  a  from  the  point  of 
starting.  Find  when  and  where  it  will  strike  the  horizontal  plane  drawn  through 
its  initial  position. 

2  V  sin  o 

Ans.  T= .     The  distance  from  the  wall  at  which  it  will  strike 

ff 

the  ground  =  e  ( o  J  ,  where  e  is  the  coefficient  of  restitution  for  the 

particle  and  the  wall. 

21.  A  large  number  of  equal  particles  are  fastened  at  unequal  intervals  to  a 
fine  string,  and  then  collected  into  a  heap  at  the  edge  of  a  smooth  horizontal 
table,  with  the  extreme  one  just  hanging  over  the  edge.  The  intervals  are  such 
that  the  times  between  successive  particles  being  carried  over  the  edge  are  equal : 
prove  that  if  cH  be  the  interval  between  the  nih  and  tbe  (n  +  l)th  particle,  and 

vn  the  velocity  just  after  the  (n  +  l)th  particle  is  carried  over,  then  —  =  —  =  n. 

Ci         Vi 

Professor  Wolstenholme,  Educ.  Times. 
If  v  be  the  velocity  acquired  by  the  first  particle  during  its  fall  through  the 
interval  ci,  we  get  immediately,  from  the  conditions  of  the  problem,  the  two  series 
of  relations 

vi  =  §0,     r2  =  §  («j  +  v)  =  f  v,    vs  =  |  (i'2  +  v)  =  %v,  &c. 

2gcx  =  v\     2gc2  =  (n  +  vf  -  vj  =  2v2,     2gcz  =  (r2  +  vf  -  v?  =  3v2,  &c, 
Hence 

vi :  vt :  Vs :  &c. :  vn  =  a :  e%\  c* :  &c. :  Cn  =  1 :  2 :  3 :  &c. :  n. 
G  2 


84 


Circular  Motion. 


CHAPTER    V. 


CIRCULAR   MOTION. 

Section  I. — Harmonic  Motion. 

87.  Uniform  Circular  Motion. — If  a  point  P  describe  a 
circle  with  a  uniform  motion,  the  radius  of  the  circle  is  called 
the  amplitude  of  the  motion,  and  the  time  of  making  one 
revolution  is  called  its  period.  If  the  arcs  are  measured  from 
a  fixed  point  A,  and  the  time  counted  from  the  instant  the 
moving  point  passed  through  a  fixed  point  E,  then  the  angle 
A  OE  is  called  the  angle  of  epoch,  or  briefly,  the  epoch.  Also 
the  ratio  which  the  arc  PE,  at  any  instant,  bears  to  the  cir- 
cumference of  the  circle  is  called  the  phase  of  the  moving  point 
at  that  instant. 

The  arrowheads  on  the  figure  denote  the  direction  in  which 
the  motion  is  supposed  to  take  place, 
and  such  a  rotation  as  there  repre- 
sented, i.e.  in  the  opposite  direction  to 
that  of  the  hands  of  a  clock,  is  con- 
sidered a  positive  rotation  :  that  in  the 
opposite  direction,  or  clockwise,  being 
considered  negative. 

Let  to  be  the  angular  velocity  of 
P,  or  the  circular  measure  of  the  arc 
described  in  one  second,  e  the  circular 
measure  of  the  epoch  AOE,  and  0  that 
of  AOP,  we  have 

0  =  wt  +  e.  (1) 

Again,  if  T  denote  the  period,  we  get  u  =  —,  and  hence, 
if  desirable,  we  should  write 

but  we  shall  generally  employ  the  form  6  =  wt  +  c,  being 
more  compendious. 


Harmonic  Motion.  85 

88.  Harmonic  .notion. — If  PM  be  drawn  perpendi- 
cular to  the  diameter  AA\  then  as  P  moves  uniformly  round 
the  circle,  the  point  M  moves  backwards  and  forwards  along 
the  line  AA\  and  is  said  to  have  a  simple  harmonic  motion. 
The  amplitude,  period,  epoch,  and  phase  of  the  harmonic 
motion  are  the  same  as  those  of  the  corresponding  circular 
motion. 

If  021  =  x,  then  the  position  of  M  at  any  instant  is  given 

by  the  equation  x  =  a  cos  (iut  +  a),  (2) 

where  a  represents  the  amplitude,  and  e  the  epoch  of  the 
motion.  The  angle  wt  +  e  is  called  the  argument  of  the 
motion,  and  the  distance  x  is  said  to  be  a  simple  harmonic 
function  of  the  time. 

Again,  if  PN  be  perpendicular  to  OB,  and  y  =  ON,  we 
have  y  =  a  sin  (cot  +  e)  =  a  cos  (tot  +  e  -  §7r). 

Hence  the  point  iV  has  also  a  harmonic  motion,  and  we 
infer  that  a  uniform  circular  motion  is  equivalent  to  two 
simultaneous  rectangular  harmonic  motions,  of  the  same 
amplitude  and  period,  but  differing  one-fourth  in  phase : 
and  conversely. 

Again,  if  the  point  M  be  projected  on  any  line,  the  pro- 
jected point  plainly  has  a  harmonic  motion  of  the  same 
period  and  phase,  but  having  for  amplitude  the  projection 
of  the  amplitude  of  M. 

If  we  differentiate  equation  (2)  we  get 

clx  . 

v  =  —  =  -  aw  sin  [wt  +  e). 

do 

Consequently  the  velocity  of  a  point  which  has  a  simple  har- 
monic motion  is  a  simple  harmonic  function  of  the  time  ;  and 
its  maximum  value  is  equal  to  the  velocity  in  the  circle. 
Again,  the  acceleration /is  given  by  the  equation 

j.    dv  „  . 

/  =  —  =  -  u)~  a  cos  (iut  +  t )  =  -  w'x. 
dt 

Consequently  the  acceleration  at  any  instant  is  propor- 
tional to  the  distance  from  the  middle  point  of  the  motion, 
and  is  always  directed  towards  that  point.  The  acceleration 
at  either  extremity  of  the  motion  is  -  u>2a. 


86  Circular  Motion. 

Any  number  of  harmonic  motions  of  equal  periods  in  the 
same  line  are  equivalent  to  a  single  harmonic  motion. 

For  let    x  =  a  cos  (iot  +  t)  +  a  cos  (iot  +  e')  +  &c. 
Then  x  =  A  cos  w£  -  5  sin  tot, 

where  A  =  2a  cos  e,     and     B  =  ^a  sin  e. 

Hence    x  -  C  cos  (o>£  +  7),     where 

C  =  v^2  +  By  an(i  tan  7  =  — . 

This  result  admits  also  of  a  simple  geometrical  demon- 
stration. 

89.  Elliptic  Harmonic  Motion. — If  a  circle  be  pro- 
jected orthogonally  on  any  plane  its  projection  is  an  ellipse, 
and  the  projection  of  any  point  which  moves  uniformly  on 
the  circle  is  said  to  have  an  elliptic  harmonic  motion. 

An  elliptic  harmonic  motion  may  be  resolved  into  two 
simple  harmonic  motions,  of  the  same  period  but  differing  in 
amplitude,  along  any  two  conjugate  diameters  of  the  ellipse, 
these  motions  differing  one-fourth  in  phase.  This  follows 
immediately  from  the  property  that  rectangular  diameters 
in  the  circle  are  projected  into  conjugate  diameters  in  the 
ellipse.  Conversely,  any  two  simple  harmonic  motions,  in 
different  lines,  of  the  same  period  and  differing  one-fourth 
in  phase,  compound  an  elliptic  harmonic  motion,  having  the 
lines  for  conjugate  diameters. 

Examples. 

1.  A  point  P  describes  a  circle  with  uniform  velocity.  If  M be  its  projective 
on  any  fixed  diameter,  prove  that  the  velocity  of  M  varies  as  PM,  and  that  its 
acceleration  varies  as  OM ;   0  being  the  centre  of  the  circle. 

2.  If  two  harmonic  motions  in  the  same  line  have  equal  amplitude  (a)  and 
equal  periods,  but  different  epochs,  e,  e',  find  the  amplitude  of  tbeir  resultant 
motion.  Ans.  2a  cos  £  (e  -  e'). 

3.  If  the  difference  of  phase  in  the  last  passes  continuously  from  0  to  2tt,  find 
the  mean  value  of  the  square  of  the  amplitude  of  the  resulting  vibration. 

Ans.  2a1. 


• 


/• 


Examples.  87 

The  mean  value  is  represented  by  the  definite  integral  {Int.  Calc,  Art.  238), 

77 
8«2fT  n         , 

—    eos-(pa<p. 

T   J  0 

This  result  is  of  importance  in  the  Wave  Theory  of  Light,  as  it  shows  that  the 
intensity  of  light  is  proportional  to  the  square  of  the  amplitude  of  the  vibration 
which  constitutes  the  light. 

4.  If  two  or  more  harmonic  motions  in  different  directions  have  the  same       •"' 
periods  and  phases,  show  that  their  resultant  is  also  a  simple  harmonic  motion 

of  the  same  phase. 

5.  Prove  that  the  resultant  of  any  number  of  simple  harmonic  motions,  dif-      S 
fering  in  directions  and  phases,  but  having  the  same  period,  is  an  elliptic  har- 
monic motion. 

6.  In  elliptic  harmonic  motion  prove  that  the  areal  velocity  of  the  moving 
point,  round  the  centre,  is  constant. 

7.  Prove  that  any  simple  harmonic  motion  is  equivalent  to  two  circular 
vibrations,  in  opposite  directions. 

8.  A  horizontal  shelf  moves  vertically  with  simple  harmonic  motion,  the 
complete  period  being  one  second.  Find  the  greatest  amplitude  in  centimetres 
that  objects  resting  on  the  shelf  may  remain  in  contact  with  it  when  at  its 
highest  point :  assuming  g=  981.  Am.  24-85. 

9.  In  elliptic  harmonic  motion,  being  given  the  difference  of  phase,  and  tbe 
ratio  of  the  amplitudes  of  the  components  along  two  given  right  lines,  perpen- 
dicular to  each  other,  determine  the  position  and  the  ratio  of  the  axes  of  the  ellipse. 

If  k  be  the  ratio  of  amplitudes,  —  the  difference  of  phase,  fxthe  ratio  of  axes, 

2lT 

and  o  the  angle  made  by  the  axis  major  with  the  direction  of  greater  amplitude, 
then 

1  +  £2  -  v/l  +  24-2cos2e  +  /».-4    ;       „       2k  cos  e 

/i2= ,  ,  tan2a=- — . 

1  +  A2  +  Vl  +  2&2cos2€  +  A4  1  -  & 

10.  Show  that  two  simple  harmonic  motions,  in  rectangular  directions,  of  the 
same  epoch,  and  whose  periods  are  as  1 :  2,  compound  a  parabolic  vibration. 

In  this  case  the  motion  may  be  represented  by  the  equations 

x  =  a  cos  2ut,     y  —  b  cos  at. 

x      2y2 
Hence,  eliminating  t,  we  get     -  =  —  —  1. 

11.  In  the  same  case,  if  the  vibrations  differ  in  epoch,  show  that  the  har- 
monic motions  compound  a  curve  of  the  fourth  degree. 

The  motion  is  represented  by  the  equations 

x  =  a  cos  (2cot  -  e),     y  =  b  cos  ut. 
Hence,  eliminating  t,  we  get 


88  Circular  Motion. 


Section  II. — Centrifugal  Force. 

90.  Centrifugal  and  Centripetal  Force. — A  heavy 
particle  may  be  made  to  move  in  a  circle,  either  by  having 
it  attached  to  a  fixed  point  by  an  inextensible  string,  and 
made  to  move  on  a  plane  passing  through  the  fixed  point,  or 
by  its  being  constrained  to  move  in  a  fixed  circular  groove. 
During  the  motion  in  the  former  case  the  string  sustains  a 
strain  or  tension  :  in  the  latter,  the  moving  particle  presses 
outwardly  against  the  groove.  This  tension,  or  pressure, 
is  called  the  centrifugal  force  of  the  particle,  and  is  always 
directed  outwards  from  the  centre  of  the  circle  described. 

The  groove,  or  string,  exerts  at  the  same  instant  an  equal 

and  opposite  reaction,  inwards  on  the  particle.     This  latter  is 

called  the  centripetal  force  which  acts  on  the  particle.     If 

m  be  the  mass  of  the  particle,  and   Fits  velocity  at  any 

instant,  then,  by  Art.  25,  we  infer  that  the  centripetal  force 

V2 
is  represented  by  m  — ,  where  r  is  the  radius  of  the  circle. 

This  result  can  also  be  established  otherwise  in  the  fol- 
lowing manner.  Let  P  be  the  position  of  the 
particle  at  any  instant ;  then  if  it  were  free,  and 
acted  on  by  no  force,  it  would  move  along  the 
tangent  at  P  with  the  velocity  V,  which  it  has  at 
the  instant ;  and  at  the  end  of  the  time  At  would 
arrive  at  the  point  iV,  where  PJ¥=  VxAt:  accord- 
ingly QN  denotes  the  space  through  which  it 
has  moved,  in  the  time  At>  owing  to  the  centri- 
petal force.  This  force  is  directed  towards  the 
centre  of  the  circle,  and  may  be  regarded  as 
constant  in  magnitude  and  direction,  during  the  indefinitely 
short  time  At. 

Hence,  if /denote  its  acceleration,  we  have,  by  Art.  36, 

QN=i/(At)\ 
But,  in  the  limit,     PN*  =  2QN.PC, 
where  C  is  the  centre  of  the  circle ; 

.-.     V\Atf=2QN.PC. 


Centrifugal  and  Centripetal  Force.  89 

V2 

Hence  / .  (1) 

r 

Or,  the  centrifugal  acceleration  f  is  a  third  proportional  to  the 

radius  of  the  circle  and  the  velocity  of  the  particle. 

mV2 
The  centrifugal  force  is  accordingly  represented  by  ■ — — . 

If  it  be  required  to  calculate  the  pressure  in  pounds  due 

W 

to  the  centrifugal  force,  we  substitute  —  for  m,  and  the  pre- 

W  V2 
ceding  expression  becomes  — — . 

Since  the  centripetal  force  is  always  directed  to  the  centre 
of  the  circle,  and  is  consequently  at  right  angles  to  the  direction 
of  motion,  it  has  no  effect  in  altering  the  velocity  of  the  mov- 
ing particle.  Hence,  if  no  other  force  act  on  the  particle,  its 
velocity  will  be  constant  during  the  motion. 

Conversely,  if  a  particle  m  describe  a  circle  of  radius  r, 

with  a  uniform  velocity  F,  we  infer  that  the  resultant  of  all 

the  forces  which  act  on  it  passes  through  the  centre  of  the 

m  V2 
circle,  and  is  represented  by  . 

Again,  as  the  velocity  in  the  circle  is  uniform,  if  T  denote 
the  number  of  seconds  in  which  the  circumference  is  described, 

we  have  V=-7zr. 

Hence,  in  this  case,  we  have 

/=4*'£.  (2) 

Consequently,  in  uniform  circular  motion,  the  centrifugal 
force  varies  directly  as  the  radius  of  the  circle,  and  inversely 
as  the  square  of  the  time  of  revolution. 

Again,  if  w  be  the  angular  velocity  of  the  radius  CP,  we 

have  (jj  =  —  :  accordingly,  in  terms  of  the  angular  velocity 

and  the  distance,  we  have 

/--V.  (3) 


90  Circular  Motion. 


Examples. 

1.  Calculate  the  centripetal  acceleration  of  a  particle  which  moves  in  a 
circle  of  5  feet  radius  with  a  velocity  of  10  feet  per  second.  Ans.  20. 

2.  A  particle  performs  20  revolutions  per  minute  in  a  circle  of  1  foot  cir- 
cumference :  find  its  centrifugal  acceleration.  Ans.  -6981. 

3.  A  hody  of  1  lb.  mass  revolves,  in  a  horizontal  plane,  at  the  extremity  of 
a  string  10  feet  long,  so  as  to  make  a  complete  revolution  in  2  seconds  ;  find  the 
tension  of  the  string  in  pounds.  107r2 

9 

4.  A  railway  carriage  of  1  ton  weight  is  moving  at  the  rate  of  60  miles  an 
hour  round  a  curve  of  1210  feet  radius :  find  the  centrifugal  pressure  on  the 
rails.  Ans.  448  lbs. 

5.  In  the  last  example,  find  how  much  the  outer  rail  should  be  raised  in 
order  that  the  total  pressure  should  be  equally  distributed  between  both  the  rails, 
the  distance  between  the  rails  being  4  feet.      Ans.  9 \  inches,  approximately. 

91.  Circular  Orbits. — In  the  case  of  uniform  circular 
motion,  since  the  centrifugal  force  acting  on  the  particle  at 
each  instant  is  directed  from  the  centre  of  the  circle,  we  may 
suppose  the  particle  to  be  kept  in  its  circular  orbit  by  the 
action  of  an  attractive  force  always  directed  to  the  centre 

r 

of  the   circle,   and  whose   acceleration  is  4?r2  — .     Hence, 

if  the  magnitude  of  the  acceleration  directed  to  a  fixed 
centre  of  force  be  known,  we  can  determine  the  conditions 
that  a  particle  should  describe  a  circle,  having  the  fixed 
point  as  its  centre.  For,  if/  be  the  acceleration  caused  by 
the  central  force  at  the  distance  r  of  the  particle,  we  have 

f  =  ^.,  and  therefore  v  =  */fr.     This  determines  the  velocity 

r 
at  each  point  in  the  circle. 

Conversely,  if  the  particle  be  projected  at  the  distance  r 
from  the  centre  of  force,  at  right  angles  to  the  radius  vector, 
with  a  velocity  v  =  yV^>  it  will  proceed  to  describe  a  circle 
freely  round  the  centre  of  force. 

2-rrr 
Also,  the  time  T  of  describing  the  circle  will  be  — — ,  or 


T=  2- 


4 


Centrifugal  Force  at  Earth's  Equator.  91 

For  example,  if  the  attractive  force  be  directly  propor- 
tional to  the  distance,  we  have  /  =  fir,  where  ju  is  some  con- 
stant ;  and  consequently,  in  this  case, 

T-2±  (4) 

Hence,  for  this  law  of  attraction  the  time  of  revolution 
in  a  circular  orbit  is  independent  of  the  distance ;  and  we 
infer  that  the  times  of  revolution  for  all  circular  orbits  round 
the  same  centre  of  force  are  equal. 

Again,  let  the  attraction  vary  inversely  as  the  square  of 

the  distance  from  the  centre,  that  is  to  say  let/=  — . 

In  this  case  the  velocity  in  the  circular  orbit  is  represented 
by   /-,  and  the  time  of  revolution  by  2?r   /— .    Hence  we  see 

that  in  different  circular  orbits  round  the  same  centre  of 
force  (which  varies  as  the  inverse  square  of  the  distance),  the 
squares  of  the  periodic  times  vary  as  the  cubes  of  the  distances 
from  the  centre  of  force. 

This  establishes  Kepler's  Third  Law  for  circular  orbits. 

The  preceding  are  particular  cases  of  general  results  con- 
nected with  the  problem  of  Central  Forces,  which  will  be 
treated  of  in  a  subsequent  Chapter. 

92.  Centrifugal   Force  at  Earth's    Equator.— We 

now  proceed  to  consider  the  centrifugal  force  arising  from 
the  rotation  of  the  Earth  on  its  axis. 

Let  r  be  the  number  of  feet  in  the  Earth's  radius  ;  T  the 
number  of  seconds  in  the  time  of  a  complete  rotation  on  its 
axis ;  /  the  acceleration  due  to  centrifugal  force  at  the  Equa- 
tor :  then  we  have 


/=4 


.2     '• 


The  most  convenient  method  of  determining /is  by  compar- 
ing its  value  with  that  of  g  at  the  Equator :  thus 

f—%  (5) 

9      <JL 


92  Circular  Motion. 

Substituting  their  numerical  values  for  7r,  r,  g,  and  T,  we 
find,  to  the  nearest  integer, 

g  -  288/ 

Again,  as  the  centrifugal  force  tends  to  diminish  the  action 
of  gravity,  it  should  be  added  to  the  observed  value  of  g  to 
obtain  the  true  acceleration  due  to  the  Earth's  attraction  at 
the  Equator. 

Hence  we  have       G  =  g  +/=  289/, 

•'•    /=2?9'  (6) 

or,  the  centrifugal  force  at  the  Equator  is  the  289th  part  of 
the  Earth's  attraction  at  the  same  place,  approximately. 

It  is  easy  from  this  result  to  determine  what  the  time  of 
the  Earth's  rotation  should  be  in  order  that  bodies  should 
have  no  weight  at  the  Equator. 

For  let  T'  be  the  time  required,  then  the  corresponding 

centrifugal  acceleration  would  be  -=^- ; 

hence  £-0 -*».*£, 

T 

Accordingly,  if  the  Earth  were  to  rotate  17  times  faster  than 
it  does  bodies  would  lose  all  their  weight  at  the  Equator. 

Hence  also  we  infer  that  if  a  body  revolve  around  the 
Earth  in  a  circle,  near  its  surface,  and  subject  to  its  attrac- 
tion solely,  it  should  travel  round  the  circumference  of  its 
circular  orbit  in  the  17th  part  of  a  day. 

93.  Verification  of  the  Law  of  Attraction. — The 
last  result  can  be  applied  to  verify,  by  a  rough  calculation, 
the  fact  that  the  Moon  is  retained  in  her  orbit  by  the  attrac- 
tive force  of  the  Earth  ;  the  law  of  force  being  the  inverse 
square  of  the  distance  from  the  Earth's  centre ;  and  the 
Moon's  path  being  assumed  to  be  a  circle  having  the  centre 
of  the  Earth  as  its  centre. 


Tangential  and  Normal  Components. 


93 


For  let  R  denote  the  distance  of  the  centre  of  the  Moon 
from  the  Earth's  centre;  T  her  periodic  time  expressed  in 
days  ;  T'  that  of  a  body  revolving  round  the  Earth  close  to 
its  surface. 

Then,  by  Art.  91,  we  have 

T2 :  T'~  =  R3 :  r3.  (8) 

Now,  assuming  the  Moon's  distance  from  the  Earth's 
centre  to  be  60  times  the  Earth's  radius,  as  found  approxi- 
mately by  observation,  we  have  R  -  60  r. 

Hence  T  =  T  60f  =  602V60  ; 

but,  by  the  last  Article,  Tf  =  —  (since  the  times  T,  T  are  ex- 
pressed in  days). 
Hence 


m       60         — 


This,  when  calculated,  gives  approximately 

T=  27-3387  days. 

The  near  agreement  (within  24  minutes)  of  this  result  with 
the  mean  value  of  T  as  obtained  by  observation,  viz., 
27*3216  days,  affords  a  strong  confirmation  of  the  Law  of 
Gravitation.  When  more  accurate  values  are  substituted, 
and  all  the  circumstances  of  the  problem  taken  into  account, 
the  calculated  agrees  completely  with  the  observed  result. 

94.  Tangential  and  Normal  Components. — Let  0 
represent  any  place,  of  latitude  A,  on 
the  Earth's  surface,  supposed  spheri- 
cal: ON  the  perpendicular  drawn  from 
0  to  the  Earth's  axis,  PP. 

Then  the  centrifugal  acceleration 
at  0  is  in  the  direction  NO  pro- 
duced ;  and  its  amount  is  represented 


4tt2NO        4ttVcosX 
— -^r— ,  or — — 


;/eosA, 


where  /  denotes  the  centrifugal  acceleration  at  the  Equator. 


94  Circular  Motion. 

The  centrifugal  force  along  NO  can  be  resolved  into  two 
components ;  one  along  CO  produced,  the  other  in  the  tan- 
gential direction. 

These  are  plainly  represented  by  /cos2A,  and/cosX  sinA, 

,.     ,              -       £cos2A          6rcosAsinA    mi       „ 
respectively;    or  by  ,  and ^-^ .    The  effect  of 

the  former  is  to  diminish  the  Earth's  attraction  as  before. 

Hence  for  the  actual  value  of  g  at  any  latitude  A,  assum- 
ing the  Earth  an  exact  sphere,  we  get 

£cos2A  /        cos2A\ 

'-*— 3ST  -*(*-  289/  (9) 

This  result  has  to  be  modified  when  the  spheroidal  form  of 
the  Earth  is  taken  into  account. 

The  tangential  component  of  the  centrifugal  force  vanishes 
at  the  Equator  and  also  at  the  poles.  For  intermediate  places 
it  varies  as  sin  2A,  and  has  its  greatest  value  at  45°  latitude, 
where  it  is  equal  to  half  the  centrifugal  acceleration  at  the 
Equator. 

Examples. 

1.  Calculate  the  diminution  of  gravity  due  to  centrifugal  force  at  a  latitude 
of  45°. 

2.  Calculate  the  tangential  component  for  the  same  case. 

3.  Calculate  the  centrifugal  acceleration  at  the  equator  of  the  planet  Mer- 
cury, its  radius  heing  1570  miles,  and  time  of  revolution  24h  5m.     Ans.   -0435. 

95.  Rotation  of  a  Rigid  Body. — If  a  rigid  body  be 
conceived  to  turn  round  a  fixed  axis,  each  of  its  points  will 
describe  a  circle,  having  its  centre  on  the  axis  of  revolution. 
Also,  since  every  line  in  the  body  that  is  perpendicular  to  the 
axis  turns  through  the  same  angle,  the  angular  velocity  of 
each  point  of  the  rigid  body  will  be  the  same  at  any  instant. 
This  instantaneous  angular  velocity  is  called  the  angular 
velocity  of  the  body,  and  it  is  plainly  the  same  as  the  velocity 
of  any  point  in  the  body  which  is  at  the  unit  of  distance  from 
the  fixed  axis. 

If  the  angular  velocity  at  any  instant  be  represented  by 
<t>,  the  velocity  of  a  point  whose  distance  from  the  axis  is  p  is 
represented  by  pw. 


Rotation  of  a  Rigid  Body.  95 

96.  If  a  plane  lamina  rotate  about  a  fixed  axis  at  right 
angles  to  its  plane,  the  centrifugal  forces  of  the  different  elements 
of  the  lamina  are  equivalent  to  a  single  force,  passing  through  its 
centre  of  mass,  and  which  is  the 
same  as  if  the  entire  mass  were  con- 
centrated at  that  po int. 

Let  the  plane  of  the  paper 
represent  that  of  the  lamina ;  and 
take  the  point  0,  in  which  the 
fixed  axis  meets  the  plane,  as  the 
origin  of  a  pair  of  fixed  rectangu- 
lar axes  OX  and  0  Y. 

Suppose  w  to  be  the  angular  velocity  of  the  lamina  at 
any  instant ;  then,  since  each  point  in  the  lamina  describes 
a  circle  round  0,  the  centrifugal  forces  for  all  its  elements 
pass  through  that  point ;  these  forces  accordingly  are  equiva- 
lent to  a  single  force.  To  find  the  value  of  this  single 
resultant;  let  OP  =  r,  and  let  dm  denote  the  mass  of  an 
element  at  the  point  P;  then  the  centrifugal  force  of  the 
element  is  wPrdm,  acting  along  the  line  OP  produced.  This 
force  can  be  decomposed  into  two,  ufxclm  and  o>2  y  dm,  parallel 
to  the  axes  of  x  and  y  respectively. 

Suppose  the  centrifugal  forces  of  the  other  elements  re- 
solved in  like  manner,  then  the  entire  system  is  equivalent 
to  the  forces  u>2  ^x  dm  and  o2  2y  dm,  parallel  to  OX  and  0  Y. 
But 

2#  dm  =  Mx,     2y  dm  =  My, 

where  x,  y  are  the  coordinates  of  the  centre  of  mass  of  the 
lamina. 

Hence  the  resultant  of  the  entire  system  of  centrifugal 
forces  is  the  same  as  that  of  the  two  forces 

w~  Mx     and     id2  My  ; 

or  to  the  single  force  w2  Md,  where  d  denotes  the  distance  of 
the  centre  of  mass  of  the  lamina  from  the  fixed  axis. 

97.  A  similar  theorem  holds  for  any  uniform  rigid  body 
turning  round  a  fixed  axis,  provided  the  body  has  a  plane  of 
symmetry  passing  through  the  axis. 

For  the  body  may  be  conceived  divided  into  a  number  of 


96 


Circular  Motion. 


indefinitely  thin  parallel  plates,  by  planes  perpendicular  to 
the  fixed  axis,  and  the  preceding  theorem  holds  for  each  of 
these  plates  or  laminae. 

Again,  if  mi9  m2,  m3,  .  .  .  &c,  denote  the  masses  of  the 
plates;  and^i,  p2,  Pz>  .  . .  &c.,  the  distances  of  their  respective 
centres  of  mass  from  the  fixed  axis;  then  as  the  body  is 
supposed  uniform  and  symmetrical,  the  centres  of  mass  of  each 
of  the  plates  all  lie  in  the  plane  of  symmetry  ;  therefore  the 
forces  w8mipi,  h>2m2p2,  . . .  &c,  form  a  system  of  parallel  forces. 
They  consequently  have  a  single  resultant,  equal  to  their 
sum,  or  to  to2Md,  where  IT  denotes  the  mass  of  the  body,  and 
d  the  distance  of  its  centre  of  mass  from  the  axis  of  rotation. 

Hence,  the  stress  on  the  fixed  axis  produced  by  the  cen- 
trifugal force  is  in  this  case  represented  by  w2Md. 

If  the  fixed  axis  pass  through  the  centre  of  mass,  the 
stress  on  the  axis  becomes  a  momental  stress  or  a  couple  :  as 
will  be  shown  also  in  the  following  Article. 

98.  Centrifugal  Forces  arising  from  Rotation  for 
a  Rigid  Body. — Next  let  us  consider  the  case  of  any  rigid 


Suppose  a  plane  drawn 
y 


body  rotating  round  a  fixed  axis, 
through  the  centre  of  mass,  per- 
pendicular to  the  fixed  axis,  and 
meeting  it  in  the  point  0.  Take 
0  as  the  origin,  the  fixed  axis  as 
that  of  z,  and  two  rectangular  axes 
as  those  of  x  and  y}  respectively. 
Let  dm  denote  an  element  of 
mass  at  the  point  P,  whose  co- 
ordinates are  x,  y,  z  ;  then,  by 
Art.  96,  the  centrifugal  force  for 
the  element  dm  is  equivalent  to 
the  forces,  io2xdm  and  w2ydm, 
acting  at  P,  parallel  to  OX  and  z 
OY,  respectively. 

Transferring  these  forces  to  the  point  N,  they  are  equiva- 
lent to  the  forces  w2xdm,  ufgdm,  parallel  to  OX  and  OF; 
along  with  the  couples  w2xzdm,  ufyzdm,  parallel  to  the  planes 
XZ  and  YZ,  respectively.  The  resultant  of  the  forces 
ufxdm,  w2ydm,  acting  at  N,  is  obviously  directed  to  0 ;  and 


Rotation  of  a  Rigid  Body.  97 

consequently  if  it  be  transferred  to  0,  it  can  be  resolved  into 
ufxdm  and  urydm,  acting  along  OiTand  OY,  respectively. 

If  each  centrifugal  force  be  resolved  in  like  manner,  the 
whole  system  is  equivalent  to  the  forces 

hflLxdm     and     uf'Eydm, 
or  to  u>~Mx        and     w2My, 

acting  along  OX  and  OY,  respectively;  together  with  the 
couples  w^xzdm,  ufSyzdm,  acting  in  the  planes  of  XZ  and 
YZ,  respectively. 

If  the  fixed  axis  be  a  principal  axis  relative  to  the  point  0 
(Int.  Cede,  Art.  214),  we  have 

*2xzdm  =  0,     and     *2yzdm  =  0. 

Hence,  in  this  case  the  strain  on  the  axis  produced  by  the 
rotation  is  the  same  as  if  the  entire  mass  was  concentrated  at 
the  centre  of  mass  of  the  rigid  body. 

If,  further,  the  fixed  axis  be  one  of  the  principal  axes 
passing  through  the  centre  of  mass  of  the  body,  the  cen- 
trifugal forces  arising  from  the  rotation  produce  no  strain  on 
the  fixed  axis.  And  accordingly,  if,  from  any  cause,  a  rigid 
body  commence  to  rotate  about  such  an  axis,  it  will  continue 
to  rotate  permanently  round  the  axis,  provided  the  only 
external  force  be  that  of  gravity. 

For  example,  if  we  suppose  a  homogeneous  sphere,  whose 
centre  is  fixed,  to  receive  any  impulse,  it  will  commence  to 
rotate  around  some  one  of  its  diameters  ;  and,  as  every  dia- 
meter is  in  this  case  a  principal  axis,  it  follows,  from  the 
preceding,  that  it  will  continue  to  revolve  permanently  round 
that  axis,  if  we  suppose  no  external  force  but  gravity  to  act 
on  it. 

On  account  of  the  property  established  above,  it  is  of  im- 
portance, in  order  that  any  machine  should  work  smoothly, 
that  the  centre  of  mass  of  any  wheel,  or  portion  which  ro- 
tates rapidly,  should  lie  on  the  axis  of  rotation,  which  should 
be  a  principal  axis  ;  for  otherwise  the  centrifugal  forces  would 
cause  strong  disturbing  vibrations. 

The  theorems  of  this  section  are  particular  cases  of  im- 
portant general  results,  which  will  be  discussed  in  a  subse- 
quent chapter. 

H 


98 


Circular  Motion. 


V 


Examples. 

1.  A  string  of  5  feet  length  is  just  capable  of  supporting  a  weight  of  10  lbs.  ; 
find  the  greatest  number  of  revolutions  per  minute  that  a  weight  of  4  lbs.  attached 
to  the  extremity  of  the  string  is  capable  of  making  in  a  horizontal  plane  without 
breaking  the  string.  Ans.  38. 

2.  A  mass  of  81bs.  is  suspended  from  the  extremity  of  a  string  10  feet  long  : 
find  the  least  velocity  that  should  be  given  to  it  in  order  to  break  the  string,  if 
its  breaking  tension  be  12  lbs.  Ans.  12 "64  feet  per  second. 

3.  Two  balls  weighing  6  lbs.  each  are  fixed  at  the  extremities  of  a  rod  of 
10  feet  length,  which  revolves  100  times  in  a  minute  around  a  central  vertical 
axis;  find  the  tension  of  the  connecting  rod.  Ans.  102 lbs. 

4.  If  two  equal  bodies  moving  on  a  rough  horizontal  plane  be  connected  by 
a  string  of  invariable  length  «,  but  without  weight ;  find  the  longest  time- that 
one  can  continue  to  move  after  the  other  has  been  stopped  by  friction. 


Ans. 


>U' 


where  p  is  the  coefficient  of  friction. 


5.  A  bodym  sliding  on  a  perfectly  smooth  horizontal  table  is  connected  by  a 
string  passing  through  a  smooth  hole  in  the  table,  with  another  body  m'  which 
hangs  freely  ;  find  the  condition  that  m'  should  remain  at  rest,  and  also  the  time 
of  revolution  of  m  in  its  circular  path,  supposed  of  radius  a. 


Ans.  Velocity  of  m  should  be    \m9a  ;  time  of  revolution 


Im'ga 
\~nT 


2tt   fam 

\  m'g 


6.  If  a  body,  attached  at  its  centre  of  mass  to  one  end  of  a  string  of  length 
ry  the  other  end  of  which  is  attached  to  a  fixed  point  on  a  smooth  horizontal 
plane,  makes  n  revolutions  per  second  ;  prove  that  the  tension  of  the  string  is  to 
the  pressure  on  the  plane  as  Air2  n-r  to  g. 

Prove  that  at  the  Equator  a  shot  fired  westward,  with  velocity  8333,  or  east- 
ward, with  velocity  7407  metres  per  second,  will,  if  unresisted,  move  horizon- 
tally round  the  earth  in  one  hour  and  twenty  minutes,  and  one  hour  and  a-half 
respectively.  Camb.  Trip.,  1878. 

7.  A  rig^Aody  of  any  form  revolves  freely  round  an  axis  fixed  in  space: 
required  thewtiditions  under  which  the  centrifugal  forces  of  its  several  elements 
will  have — (a)  no  resultant ;  (b)  a  resultant  pair  ;  (c)  a  resultant  single  force  ; 
(d)  a  resultant  pair  and  single  force.  Lloyd  Exhib.,  1872. 

Section  III. — Motion  in  a  Vertical  Circle. 

99.  Velocity  in  a  Smooth 
Vertical  Curve Before  the  dis- 
cussion of  motion  in  a  circle  we 
shall  consider  some  properties  of 
the  motion  of  a  particle,  under  the 
action  of  gravity,  on  any  vertical 
curve. 

Take  OX  a  horizontal,  and 
07a  vertical  line  in  the  plane  as 
axes  of  coordinates ;  and  suppose 
x,  y  the  coordinates  of  P,  the  position  of  the  particle  at  the 


y 

r 

i 

p 

<y 

N 


Motion  in  a  Vertical  Circle.  99 

end  of  any  time  t.  Let  A  be  its  position  when  t  =  0,  and  let 
AP  =  s;  then,  by  Art.  43,  the  acceleration  along  the  curve  at 
P  is  represented  by 

.     .  dii 

drs  dy 


Hence  we 

have 

~df=~gck' 

Multiply  by 

2ds, 

and  integrate ; 

then 

r 

■$--*♦ 

const. 

Let  f/n 

=  AB, 

and 

«?o  =  velocity  at  ^4,  then 

2  _   _ 


'0 


2gy0  +  const. ; 


therefore  v2  -  ty  =  2g  (yQ  -  y).  (2) 

Again,  if  AR  be  measured  upwards  =  h,  the  height  due 
to  the  velocity  v0,  and  ED  be  drawn  parallel  to  the  axis  of  x ; 
then 

v>  =  2g(li  +  yQ-y)  =  2gPL.  (3) 

Consequently  the  velocity  at  any  point  P  is  the  same  as  that 
acquired  in  falling  from  the  horizontal  line  DR. 

This  is  an  extension  of  the  result  given  in  Art.  49,  and  is 
itself  a  case  of  the  general  principle  of  work  which  shall  be 
treated  of  in  the  next  chapter  (sec  Art.  132). 

100.  Motion  in  a  Vertical  Circle. — If  a  particle  be 
constrained  to  move  in  a  vertical  circle  under  the  action  of 
gravity,  its  velocity  at  any  point,  by  (2) ,  is  the  velocity  due  to 
falling  through  a  certain  height  from  a  certain  horizontal  line, 
or  level.  The  motion  will  be  one  of  complete  revolution  if 
this  right  line  lies  altogether  outside  the  circle.  If  the  line 
cut  the  circle  the  motion  will  be  oscillatory.  We  proceed  to 
consider  the  latter  case  in  the  first  instance.  In  this  case 
we  may  either  consider  the  particle  as  moving  in  a  smooth 
circular  tube,  or  as  attached  by  an  inextensible  string  to  a 
fixed  point  in  the  centre  of  the  circle,  the  weight  of  the  string 
being  neglected. 

H2 


100 


Circular  Motion. 


When  the  arc  in  which  the  oscillation  has  place  is  but  a 
small  portion  of  the  circumference  we  get  what  is  called  a 
simple  pendulum.  From  this  statement  the  student  will 
see  that  a  simple  pendulum  can  only  be  approximately  repre- 
sented. However,  a  small  leaden  ball  suspended  from  a, 
fixed  point  by  a  very  fine  wire  may  be  regarded  approxi- 
mately as  a  simple  pendulum. 

101.  Simple  Pendulum. — Let  C  be  the  centre  of  the 
circle ;  0  its  lowest  point ;  A  the  point 
from  which  the  particle  may  be  sup- 
posed to  start ;  P  its  position  at  the 
end  of  any  time  t ;  v  the  corresponding 
velocity, 

Q=LPCO,     a=LACO, 
estimated  in  circular  measure, 
s  =  AP,     Z=OC. 

Then,  since  the  velocity  at  P  is  that  due  to  falling  from 
a  horizontal  line  drawn  through  A,  we  have 

v2  =  2gl  (cos  0  -  cos  a) ; 
It. 


but 


#  =   - 


\dt 


therefore 


=  ^(cos0 


COS  a) 


i 


sin' 


■n-g. 


Consequently      ^  =  ±  2  ^/f  V sin^ 


~  -  sin* 


2' 


Again,  since  in  the  motion  from  A  to  0,  6  diminishes  as 
t  increases,  —  is  negative.     Accordingly  we  have 

ut 


§-4MM- 


Time  of  a  Small  Oscillation.  101 

l~  CO  "         7fi'  ' '    '     "'  j    '    -  "'  •   *   ' 

Hence  we  get       tf    / f  =  -      — -===-.  (4) 

102.  Time  of  a  Small  Oscillation. — The  preceding 
definite  integral,  which  represents  the  time  of  describing  a 
circular  arc,  cannot  be  expressed  in  finite  terms  by  means  of 
the  ordinary  algebraic  or  trigonometrical  functions ;  however, 
when  the  amplitude  of  the  oscillation  is  small  we  can  easily 
get  an  approximate  value  for  t,  as  follows : — When  a  is  so 
small,  that  we  may  neglect  powers  of  a  and  9  beyond  the 
second,  we  have 

4(Wg-sin2^  =  a2-0\ 


Hence  (4)  becomes 


?< 


do  jo 

cos"1 


/a2  -  02  \« 


No  constant  is  added  as  0  =  a  when  t  =  0.     Consequently 
we  have 


acosjf 


0  =  a  COS    /|  *.  (5) 

Accordingly  0  is  a  simple  harmonic  function  of  the  time 
(Art.  88). 

Again,  when  6  =  0,  we  have  Jj  t  =  - ;  hence  the  time  of 

descent  to  the  lowest  point  is  represented  by  o  J~- 

The  particle,  after  arriving  at  the  lowest  point,  plainly 
moves  up  the  other  side  of  the  arc,  and  if  the  whole  time  of 
a  small  oscillation,  expressed  in  seconds,  be  denoted  by  T,  we 
have 

r-w£  (6) 

Since  this  expression  is  independent  of  a,  it  follows  that 


102  -Circular  Motion. 

the  time '  cf'ia  smrttt  oscillation is  the  same  for  all  arcs  of  vibra- 
tion in  the  same  circle.-  From  this  property  the  vibrations  of 
a  pendulum  are  said  to  be  isochronous.  Also  the  time  of  a 
small  oscillation  at  any  place  varies  as  the  square  root  of  the 
length  of  the  pendulum. 

103.  The  Seconds  Pendulum. — A  pendulum  which 
oscillates  once  in  every  second  is  called  a  seconds  pendulum. 
If  L  be  its  length,  since  the  corresponding  value  of  T  is 
unity,  we  have 

g  =  7T2X.  (7) 

Hence  the  value  of  g  can  be  determined  for  any  place 
whenever  the  corresponding  value  of  L  is  obtained. 

This  gives  the  most  accurate  method  of  finding  the  value 
of  g  at  any  place,  since  that  of  L  can  be  determined  with 
great  accuracy  by  observation. 

Any  rigid  body  made  to  vibrate  about  a  fixed  horizontal 
axis  is  called  a  compound  pendulum .  It  will  be  shown  sub- 
sequently {see  Art.  135)  that  in  every  such  case  there  is  an 
equivalent  simple  pendulum  which  would  vibrate  in  the  same 
time  as  the  actual  pendulum  under  consideration.  This  cir- 
cumstance renders  the  consideration  of  the  ideal  pendulum 
above  discussed  of  the  utmost  practical  importance. 

The  length  of  a  seconds  pendulum  at  London  is  found  to 
be  39*1416  inches,  approximately;  hence  the  corresponding 
value  of  g  is  32*1908  feet. 

Pendulum  observations  furnish  the  most  accurate  proof  of 
the  fact  that  the  force  of  gravity  acts  with  equal  intensity 
on  all  substances,  as  it  will  be  seen  that  the  length  of  the 
simple  pendulum  equivalent  to  any  compound  one  .depends 
merely  on  the  shape  of  the  latter,  but  not  on  its  material, 
provided  it  be  homogeneous. 

Again,  if  T,  T'  be  the  times  of  small  oscillation  for  two 
pendulums  of  different  lengths,  /  and  V;  and  if  n  and  ri  be 
the  number  of  their  respective  vibrations  in  the  same  time 
(a  day  suppose),  we  shall  have 

•-■^-    P.  (8) 

Hence,  if  the  length  I  of  any  simple  pendulum  be  known, 


Change  of  Length.  103 

and  also  the  number  n  of  its  vibrations  in  a  day,  the  length 
L  of  the  seconds  pendulum  at  the  place  can  be  calculated. 
For,  since  the  number  of  seconds  in  a  day  is  86400,  we  have, 
from  formula  (8), 

H«)2/-  (9) 

The  time  T  of  vibration  of  a  pendulum  varies  either— 
(1)  by  altering  the  length  I  of  the  pendulum,  or  (2)  by 
changing  the  place  of  vibration.  We  shall  consider  these 
causes  independently. 

104.  Change  of  Length. — Adopting  the  same  notation 
as  before,  we  get 

*L  -  r 
w'2  "  I  ; 

n2  -  n2       V  -  I  ,        n2      V  -  I 

hence  T, —  =  — r~  5    •'•  n  ~  n  = ?  — T~- 

n-  I  n+ n        I 

When  the  change  in  length  is  a  very  small  fraction  of 
the  whole  length,  n  and  n  are  nearly  equal,  and  we  have, 
approximately, 


n  +  n      2 
Accordingly,  in  this  case, 

n-n=-T;  (10) 

where  £l  denotes  the  change  of  length  of  the  pendulum. 

If  the  pendulum  be  lengthened,  i.e.  if  Al  be  positive, 
n  -  n'  is  positive,  and  hence  the  number  of  vibrations  in  a 
given  time  is  diminished  when  the  length  of  the  pendulum  is 
increased,  as  is  otherwise  evident. 

In  the  case  of  a  seconds  pendulum  we  substitute  L  for  J 
in  the  preceding  ;  and  since  n  =  86400,  we  get  for  the  dimi- 
nution in  the  number  of  vibrations  in  a  day, 

43200  ^. 

JU 


104  Circular  Motion. 

Hence  we  can  determine  the  number  of  seconds  gained 
or  lost  by  a  seconds  pendulum  in  a  day  when  its  length  is 
slightly  altered. 

As  bodies  in  general  expand  slightly  with  an  increase  of 
temperature,  an  ordinary  clock  should  go  slower  in  hot 
weather,  and  faster  in  cold.  The  different  methods  of 
compensation  for  correcting  the  error  arising  from  this 
cause  will  be  found  in  practical  treatises  on  the  subject. 
The  amount  of  expansion  for  an  increase  of  temperature  for 
different  substances  has  been  accurately  determined,  and 
registered  in  Tables. 

If  AL  denote  the  change  in  the  length  of  a  seconds  pen- 
dulum arising  from  this  cause,  the  corresponding  loss  or  gain 
can  be  determined  by  (10). 

We  add  a  few  examples  for  illustration. 

Examples. 

1 .  Calculate  the  length  of  a  pendulum  beating  seconds  in  London,  assuming 
^  =  32-19. 

2.  If  the  bob  of  a  seconds  pendulum  be  screwed  up  one  turn,  the  serew 
being  32  threads  to  the  inch  ;  find  the  number  of  seconds  it  should  gain  in  the 
day  in  consequence,  assuming  L  =  39'14  inches.  Ans.  34'7  seconds. 

3.  A  heavy  ball,  suspended  by  a  fine  wire,  makes  885  oscillations  in  an 
hour.  Find  the  length  of  the  wire  approximately,  assuming  the  length  of  the 
seconds  pendulum  to  be  39' 14  inches.  Am.  54  feet. 

4.  Find  the  error  in  one  day  produced  by  an  increase  of  15°  F.  of  tempera- 

Al  1 

ture  in  a  steel  seconds  pendulum ;  assuming  that  —  for  10°  F.  =  . 

Ans.   4*15  seconds. 

5.  A  seconds  pendulum  is  lengthened  -r0-th  of  an  inch ;  find  the  number  of 
seconds  it  will  lose  in  one  day.  Ans.  110*4. 

105.  Change  of  Place. — The  acceleration  g  varies* 
from  place  to  place,  and  consequently  the  number  of  vibra- 

*  For  places  at  the  sea  level,  this  arises  from  two  causes— one,  the  variation 
of  centrifugal  force  already  considered  (Art.  94)  ;  the  other,  that  the  Earth  is 
not  an  exact  sphere,  but  is  more  nearly  an  oblate  spheroid  of  revolution  round 
its  axis  of  rotation.  From  each  of  these  it  arises  that  the  value  of  g  diminishes 
in  proceeding  from  the  pole  to  the  equator.  It  was  from  the  observation  by 
Richer,  in  1672,  that  a  clock  lost  two  minutes  daily  when  taken  to  Cayenne, 
lat.  5°  N.,  and  that  when  the  corrected  pendulum  was  brought  back  to  Paris 
it  gained  an  equal  amount,  that  the  variation  of  the  force  of  gravity  on  the 
Earth's  surface  was  first  established.     The  explanation  is  due  to  Huygens. 


Change  of  Place.  105 

tions  in  a  given  time  will  vary  with  the  place  for  the  same 
pendulum. 

Suppose  n  and  ri  to  represent  the  number  of  vibrations 
made  in  one  day  by  the  same  pendulum  at  two  places,  at 
which  g  and  g  are  the  corresponding  accelerations,  we  have 

?L-Lm    t     or  nl-t 
n       T      Sg'  n*      g' 

Hence,  as  before,  for  one  and  the  same  pendulum, 

From  this,  if  L  and  II  be  the  lengths  of  the  seconds 
pendulum  at  the  two  places,  we  get 

nL'-L 
n-n^-JT-  (12) 

It  is  shown  by  theory,  and  verified  by  observation,  that 
the  variation  in  the  length  of  L,  and  consequently  in  g,  at 
the  sea  level,  is  proportional  to  the  square  of  the  sine  of 
the  latitude  (compare  Art.  94).  Thus,  if  L  denote  the  length 
of  the  seconds  pendulum  at  the  equator,  II  that  at  latitude  A', 
we  have 

Z/  =  Z  +  ™sin2A'.  (13) 

Hence,  if  Lx  be  the  length  of  the  seconds  pendulum  at 
45°  latitude,  we  have  Lx  =  L  +  — . 

Eliminating  L,  we  get 

L'  =  Ll-^cos2X.  (14) 

Again,  if  L"  be  the  length  corresponding  to  the  latitude 
A",  and  g"  the  corresponding  value  of  g,  we  have 

m-f  (cos2A"-cos2r) 


ST         2L' 


—  sin  (A'  +  X')  sin  (A'  -  A"),  approximately. 


106  Circular  Motion. 

By  accurate  observation  of  the  number  of  vibrations  lost 
by  a  pendulum  which  beats  seconds  at  the  latitude  A,  when 

111 
taken  to  a  latitude  A',  the  value  of  y-  can  be  determined. 

Such  observations  give  -=-   =  zr^=,   approximately,   and 

Zi=  39-118  inches. 
Hence,  we  get 

If  =39-118-^  cos  2\'. 

Again,  suppose  a  pendulum,  beating  seconds  at  any 
place,  taken  to  the  height  h  above  the  Earth's  surface  at  that 
place  ;  and  let  g  be  the  value  of  g  for  the  new  position ; 
then,  since  the  force  of  gravity  varies  as  the  inverse  square 
of  the  distance  from  the  Earth's  centre,  we  have 

g'  =  9  (r  +  hf  "  * ( *  ~  7/  aPProximately> 
where  r  denotes  the  length  of  the  Earth's  radius ;  therefore 

g  -  g  =    2A 

9  r* 

Hence,  when  -  is  a  very  small  fraction,  the  number  of 

seconds  lost  in  a  day  by  the  seconds  pendulum  is  86400  -. 

Suppose,  for  example,  h  =  1  mile,  and  r  =  3956  miles, 
then  the  number  of  seconds  lost  in  a  day  will  be  22,  approxi- 
mately. 

In  this  investigation  the  attraction  on  the  pendulum  of 
the  part  of  the  Earth  above  the  sea  level  has  been  neglected. 

Examples. 

1.  If  a  pendulum,  beating  seconds  at  the  foot  of  a  mountain,  lose  10  seconds 
in  a  day  when  taken  to  its  summit;  find  approximately  the  height  of  the 

^1  mountain,   assuming  the  radius  of  the  Earth  4000  miles,  and  neglecting  the 

^         attraction  of  the  mountain.  Am.  2444  feet. 

2.  How  much  would  a  clock  gain  at  the  equator  in  24  hours  if  the  length  of 
the  day  were  doubled.  Am.  112|  seconds,  approximately. 


Airyh  Investigation  of  Mean  Density  of  Earth.        107 

106.  Airy's  Investigation  of  the  Mean  Density  of 
the  Earth.— A  series  of  important  pendulum  experiments 
were  undertaken  by  Sir  Gr.  B.  Airy,  in  the  Harton  coal  mine, 
for  the  purpose  of  determining  the  mean  density  of  the  Earth. 
He  found  that  a  pendulum  beating  seconds  at  the  surface 
gained  2|  seconds  a-day  when  taken  to  the  bottom  of  the 
mine,  1260  feet  deep.  The  calculations  employed  in  arriving 
at  this  result,  and  in  determining  from  it  the  Earth's  mean 
density,  are  very  intricate  ;  they  will  be  found  in  the  Eoyal 
Society's  Transactions  for  the  year  1856. 

The  following  is  a  method  of  arriving,  approximately,  at 
the  result : — 

Let  g,  g  denote  the  accelerations  due  to  gravity  at  the 
surface  and  at  the  bottom  of  the  mine  ;  then,  by  equation  (11), 
we  have 

g  43200      19200 

Again,  let  r  and  r  denote  the  distances  of  the  upper 
and  lower  stations  from  the  centre  of  the  Earth,  supposed 
spherical. 

Suppose  a  concentric  sphere  described  through  the  lower 
station,  then  the  attraction  of  the  Earth  at  the  upper  station 
may  be  regarded  as  consisting  of  two  parts — one  due  to  the 
interior  sphere,  the  other  to  the  concJw  or  shell,  bounded  by 
the  two  spheres.  Again,  if  we  suppose  this  shell  to  be  of 
uniform  density,  it  exercises  no  attraction  on  the  pendulum 
at  the  bottom  of  the  mine.  This  can  be  easily ^  seen  from 
elementary  geometrical  considerations  (Minchin,  Statics, 
Art.  319).  Hence  the  part  of  g  due  to  the  attraction  of  the 
inner  sphere  is  represented  by 

/o 

o  7r 

If /denote  the  acceleration  at  the  upper  station  due  to 
the  attraction  of  the  shell,  we  have 

<7=/+^=/+</(l +  19300)7- 
Again,  let  h  represent  the  depth  of  the  mine,  then  r=  r  -  h  ; 


108  Circular  Motion. 

and,  since  -  is  very  small,  we  have  -7=1 7,  approxi- 
mately. 

Accordingly  we  get,  from  the  preceding  equation, 

,    m     1  \ 

In  order  to  get  another  expression  for/,  let  M,  V,  B  de- 
note respectively  the  mass,  volume,  and  mean  density  of  the 
Earth ;  and  m,  v,  p  the  corresponding  quantities  for  the  shell. 
"We  assume  that  the  Earth  and  the  shell  each  attract  as  if 
their  whole  mass  was  concentrated  at  their  common  centre ; 
in  this  case  we  have,  approximately, 

*      m         p  v         o  rz  -  r'z      _     p  h 

XT  2h  1  „hP 

HenCe  7-19200  =3,-£- 

Substituting  3956  miles  for  r,  and  1260  feet  for  h,  we  get 
D  =  2-625,0.  (15) 

The  determination  of  the  mean  density  of  the  Earth  is 
thus  reduced  to  finding  the  value  of  p  ;  but  this  is  a  matter 
of  extreme  practical  difficulty. 

From  an  accurate  examination  of  the  mineral  components 
of  the  stratum  of  the  Earth  in  the  neighbourhood  of  the 
mine,  p  was  calculated  by  Airy  to  be  2  J  times  the  density  of 
water.  This  would  give  the  mean  density  of  the  Earth 
about  6-66,  assuming  that  of  water  as  unity. 

Professor  Haughton  calculated  2*059  as  the  value  of  p 
(Phil.  Trans.,  July,  1856),  adopting  as  his  basis  Humboldt's 
investigations  of  the  mean  heights  of  Continents  on  the  Earth's 
surface,  and  Bigault's,  on  the  relative  areas  of  land  and 
water.  This  would  give  5*405  as  the  value  of  the  mean 
density  of  the  Earth. 

107.  Time  of  Oscillation  in  General. — The  ampli- 
tude of  the  vibration  has  hitherto  been  considered  so  small 
that'powers  of  a  higher  than  the  second  have  been  neglected. 


Time  of  Oscillation  in  General.  10£> 

We  now  proceed  to  find  a  general  expression  for  the  time  T 
of  vibrations  for  any  amplitude. 

From  (4),  since  \T represents  the  time  to  the  lowest  point 
on  the  circle,  we  get 


'■M 


Jsin"rsm2 

Now,  assuming*  sin  -  =  sin-  sin  <£,  we  get 
dd  2fty  2<fy 


(16) 


^Jsin2^  -  sm2^      cos^     ^1  -  sin--  snrtf> 
Also,  when  0  =  0  we  have  0  =  0;  and  when  0  =  a  we  have 

7T 

=  2* 


Consequently        21  =  2 J-  P  ^ 


(it; 


2  sm-  ^ 

Again,  substitute  A;2  for  sin2  -  ;  then,  since 

(1  -  A<2  sin2  tj>)~*  =  1  +  \  k2  sin2tf>  +  ^|  ¥  sin4tf> 
1.3.5 


2.4.6 
and  (Jw*.  Cfcfc.,  Art.  93), 


7t6sin6</>  +  &c, 
1.3.5...(2m-l)ir 


r2  l . 

J0         r  r    2.4.6...     2m       2 
we  get 


r-4.m*t<m» 


OT6)V+to-i 


*  This  assumption  is  obviously  a  legitimate  one,  because  0  during  the  motion 
can  never  be  greater  than  a. 


110  Circular  Motion. 

If  h  be  the  vertical  height  of  the  point  from  which  the 
pendulum  commences  to  descend  above  the  lowest  point  in 
the  circle,  we  have 

.     n      1  -cos  a      h 
j..mi_  ______  ; 

and  the  preceding  result  becomes 


(19) 


liU  AV*   /i.3\yav  /i-3.5\y/A3 
r-N,|1+(?)5i+Wfe)+(oi)l5i)+-" 

The  first  term  gives  the  value  already  arrived  at  for  a 
small  oscillation,  and  is  independent  of  the  amplitude. 

The  second  approximation,  which  is  the  one  commonly 
adopted  when  the  lowest  powers  of  the  amplitude  are  taken 
into  account,  gives 


(21) 


ff\      8;/ 
In  terms  of  the  semi-amplitude  a,  this  is 


-4K 


in  which  a  is  taken  in  circular  measure. 

The  general  equation  (16)  is  immediately  integrable  in 
one  case,  viz.,  when  the  velocity  at  any  point  on  the  circle  is 
that  due  to  a  fall  from  its  highest  point ;  for  in  this  case  we 

have  a  =  7T,  and  therefore  sin  |  =  1.     Equation  (4)  becomes 

in  this  case 

dO 


-afe*5 


cosJ0 
hence  we  get 

2i^=  log  tan  I  (it  +  0)  +  const. 

It  may  be  observed  that  in  this  case,  since  log  0  =  -  oo  , 
the  particle  would  take  an  infinite  time  to  reach  the  highest 
point  on  the  circle. 


d2x 
Integration  of  —  ±  fix  =  0 

clo 

Examples. 


Ill 


1.  How  is  the  value  for  the  time  of  vibration  of  a  pendulum  to  he  corrected 
when  the  length  of  the  arc  of  vibration  is  taken  into  account  ? 

2.  Apply  to  the  case  where  the  amplitude  of  vibratiou  is  120°. 

„  •  h  l 

Here,  since  — ■  =  -,  we  have 
21      4 


'-vEK*iBi+*) 


3.  If  a  pendulum,  which  heats  seconds  for  very  small  oscillations,  he  made 
to  vibrate  through  an  arc  of  10°;  find,  approximately,  the  number  of  seconds 
it  should  lose  in  a  day.  Ans.  41. 

108.  motion  in  a  Vertical  Cycloid. — Let  a  particle 
be  supposed  to  move  along  a  smooth  Y 
cycloid,  having  its  vertex  0  at  its  lowest 
point,  and  its  axis  0  Y  vertical. 

Calling    OP   =   8,   PN  ^  y,   and 
a  =  diameter  of  generating  circle. 

Then  {Biff.  Cede,  Art.  276),  we  have 


or 


s2  = 

lay. 

,  from 

(1), 

d2s 

dy 
■'5- 

9 
2a 

d2s 
df 

+  ts  = 

-0. 

(22) 


We   shall  next    consider   the   method    of  integrating   this 
equation. 

d2x 
109.  Integration  of  —  ±  fix  =  0. — As  differential  equa- 


d2x 


dt 


tions  of  the  form  -r^  ±  fix  =  0  are  of  frequent  occurrence  in 
physical  problems,  we  proceed  to  consider  their  solution. 


112  Circular  Motion. 

There  are  two  cases,  according  as  the  upper  or  lower  sign 
is  taken. 

1st.— Let  -Tp  +  fix  =  0. 

Multiplying  by  2dx9  and  integrating,  we  get 

—  J  +  \xx2  =  const. 

dx 
To  determine  the  constant,  suppose  x  =  a  when  —  =  0> 

then  the  constant  is,  plainly,  \xa2 ; 

fdx\2 
therefore  ( -j  )  =  fi(a2-  x2). 

Hence  '  7  „  =  vM^j 

y/  a-  -  x~ 

or  sin-1-  =  t^fi  +  a, 

ci 

where  a  denotes  an  arbitrary  constant. 

Consequently     x  =  a  sin  if^ii  +  a),  (23) 

where  a,  a  are  arbitrary  constants,  to  be  determined  in  each 
case  by  the  conditions  of  the  problem. 

It  may  be  observed  that  x  is  a  simple  harmonic  function 
of  the  time  (Art.  88). 

The  preceding  solution  admits  of  being  also  written  in 
the  form 

x  =  C  cos  t  ^/fx  +  C  sin  t  y/ji,  (24) 

where  C  and  C  are  two  arbitrary  constants. 

Either  of  the  latter  equations  may  be  regarded  as  the 
complete  integral  of  the  differential  equation 

d2x 


It2 


Integration  of  —^  ±  fix  =  0.  113 


2nd-  aV=fxX' 

Proceeding  as  before,  we  get 

in  which  a  is  an  arbitrary  constant ; 

dx 

or  .  =  at .  /.. . 

therefore,   Vm  +  a  =  J  /^=g  =  log  ^  +  -/^A 

in  which  a  is  arbitrary. 


Hence  #  +  */»a  -  #3  =  &■**  =  Aeu* 

where  A  is  arbitrary. 
Again,  since 


(x  +  yV  -  a2)  (x  -  */x2  -  a2)  =  a2, 

, a2     .- 

we  get  x-</x2-a2  =  2e      ' 

Adding,  we  obtain 


2x  =  Aeu*  +  —  e-t\ 


a~ 
A 


which  may  be  written  in  the  form 

x  =  Ceu»  +  <?V^>,  (25) 

when  C  and  C  are  two  arbitrary  constants,  to  be  determined, 
as  before,  by  the  conditions  of  the  problem  in  each  particular 
case. 

i 


114  Circular  Motion. 

110.  The  equation 

(Px 

_  +  ^  +  v  =  0 

is  immediately  reducible  to  the  preceding,  for  it  may  be 
written 

d2x        (       v\     A 

If  we  substitute  z  for  x  +  -,  this  becomes  ■=-=  +  jtes  =  0  ; 

jjl  ax 

consequently  we  have 

®  =  ~  ~  +  C  cos  t^Z/j.  +  C  sin  £  v  p. 
In  like  manner  the  solution  of 


d2x  n 


is  x 


=  Y.  +  Qeu»  +  C'e-U*. 


111.  Time  of  Oscillation  in  Cycloid. — Returning  to 
equation  (22),  Art.  108,  and  substituting  s  for  x,  and—  for  /*  in 
equation  (24),  we  find  for  its  integral 

.    ...  oo.  *j£ +  •*.*,/£.  (26) 

In  order  to  determine  the  constants  c  and  c',  suppose  the 
particle  to  start  from  rest,  at  the  distance  s'  from  the  ver- 
tex 0  (measured  along   the  curve)  ;  then   we  have 


s  = 


and  —  =  0,  when  t  =  0.     Making  these  substitutions  in  (26), 

dt 
as  well  as  in  the  equation  derived  from  it  by  differentiation, 

we  get. 

c  =  s%     and    c  =  0  ; 

therefore  *  =  »'  cos  t^.  (27) 


Conical  Pendulum. 


115 


Again,  when  s  =  0,  we  get 


or  t 


3 


this  gives  the  time  of  descent  to  the  lowest  point.     If  T  de- 
note the  time  of  an  oscillation,  we  have 


-V 


2a 


(28) 


Since  this  result  is  independent  of  the  length  of  the  arc  of 
vibration,  it  follows  that  the  time  of  vibration  is  the  same  for 
all  arcs  of  the  cycloid;  accordingly  the  property  of  tautochro- 
nism,  which  in  the  circle  holds  only  for  very  small  arcs,  holds 
in  all  cases  for  the  cycloid  (compare  Art.  88). 

The  foregoing  value  of  T  is  the  same  as  that  for  a  small 
oscillation  in  a  vertical  circle  of  radius  2a.  Moreover,  as  2a 
is  the  radius  of  curvature  at  the  vertex  of  the  cycloid'  {Biff. 
Calc,  Art.  276),  the  duration  of  an  oscillation  in  a  vertical 
cycloid  is  the  same  as  that  of  a  small  oscillation  in  the  circle 
which  osculates  it  at  its  lowest  point ;  as  is  manifest  also  from 
other  considerations. 

It  is  readily  seen  that  the  time  of  an  indefinitely  small 
oscillation  about  the  lowest  point  in  any  plane  vertical  curve  is 
the  same  as  that  in  the  osculating  circle  at  the  lowest  point ; 

and  its  duration  is  accordingly  represented  by  irJ-9  where  p 
denotes  the  radius  of  curvature  at  the  point. 

112.  Conical  Pendulum.— Suppose  the  pendulum,  in- 
stead of  moving  in  a  vertical  plane,  to  describe 
a  right  cone  around  a  vertical  axis;  and  let 
P  be  the  position  of  the  revolving  particle  at 
any  instant;  0  the  point  of  suspension ;  PiV'the 
perpendicular  let  fall  on  the  vertical  axis. 

Also  let  OP  =  1,     L  PON  =  0. 

Then  the  motion  of  P  may  be  considered  as   n 
taking  place  in  a  horizontal  circle,  whose  centre 
is  N,  and  radius  PiV  or  I  sin  9. 

i2 


116  Circular  Motion. 

Now,  in  order  that  this  motion  should  take  place,  it  is 
necessary  that  the  resultant  of  the  tension  of  the  string  and 
the  weight  of  the  particle  should  act  along  PN,  and  be  equal 
and  opposite  to  the  centrifugal  force ;  i.e.  that  the  resultant  of 

the  weight  W,  and  the  centrifugal  force,  —  —. — r.,  should  act 
&         '  9  IsmO 

in  the  line  OP.     This  gives 

W    v2 
W:—-r^-a=ON:PN, 

g  Ism  t) 


or 


_  .    ,PN       ;sin20 


ON    v  cos  6  ' 
therefore  v  =  sin  Bj—#  (29) 

A  COS  U 

This  gives  the  velocity  in  terms  of  6  and  /. 

Again,  if  T  be  the  time  of  revolution,  we  have 


T^PN 


^  (30) 


This  determines  the  time  of  revolution  when  the  angle  0, 
which  the  pendulum  makes  during  the  motion  with  the  ver- 
tical, is  known.  It  is  evidently  the  same  as  that  of  a  double 
oscillation  in  a  simple  pendulum  of  length  /cos0  or  ON. 
The  tension  of  the  string  is  represented  by  IFsec  6.  The 
preceding  is  a  particular  case  of  the  motion  of  a  particle  on 
a  smooth  sphere,  a  problem  which  will  be  considered  in 
Chapter  VIII.. 

113.  Watt's  Governor. — The  principal  of  the  conical 
pendulum  was  employed  by  Watt,  in  the  instrument  called  a 
governor,  for  the  purpose  of  regulating  the  supply  of  steam 
so  as  to  maintain,  approximately,  a  steady  motion  in  a  steam- 
engine.  Its  construction,  under  a  form  which  is  commonly 
employed,  is  as  follows  : — 


Revolution  in  a  Vertical  Circle. 


117 


Let  AB  represent  a  vertical  spindle  rotating  with  an 
angular  velocity,  whose  speed  is  so 
regulated  as  to  be  always  propor- 
tional to  that  of  the  machine.  CP 
and  CI*  are  rigid  rods,  jointed  at  C 
and  C  upon  the  revolving  spindle, 
and  having  massive  equal  balls,  P and 
P',  fixed  at  their  extremities.  FJD  and 
F'D'  are  two  rods  also  jointed  at  D 
and  I)f  to  the  rigid  rods,  and  jointed 
at  F  and  F'  to  a  collar,  movable 
freely  on  the  spindle.  The  collar  at 
F,  sliding  freely  up  and  down  the 
spindle,  is  united  to  a  lever  which 
opens  or  closes  the  valve  that  regu- 
lates the  supply  of  steam  to  the 
cylinder  of  the  engine.  When  the  shaft  AB  turns  too  fast, 
the  balls  P  and  P'  fly  from  it,  raising  the  collar  F,  and 
thus  diminishing  the  supply  of  steam,  and  consequently  re- 
ducing the  speed.  For  a  more  complete  discussion  the 
student  is  referred  to  works  on  practical  mechanics. 

114.  Revolution  in  a  Vertical  Circle. — We  now  re- 
turn to  the  question  of  the  revolution  of  a  particle  in  a  vertical 
circle  under  the  action  of  gravity. 

Suppose  DR  to  be  the  horizontal  line 
to  the  distance  below  which  the  velocity 
at  any  point  is  due,  and  let  AD=h;  then, 
by  Art.  99,  the  velocity  at  any  point  P 
is  given  by  the  equation 

v~=2g(h-AN)  =  2g(h-2asm2±0), 

where  PCA  =  0. 

i 

Hence,  denoting  —  by  k2,  and  substituting  a2  ( —  )  for  v\ 
we  get 


dt 


118  Circular  Motion. 

therefore  ^=-7   /Vl-Fsin^fl, 

dt         k\a  2  ' 

in  which  k  is  less  than  unity. 

If  0  =  Z.  P-B^4  =  J0,  the  time  of  describing  any  arc  of  the 

circle  is  represented  by  the  definite  integral 

(31) 


.y/l  -  /rsin2^ 

where  a  and  /3  are  the  values  of  <p  corresponding  to  the  ex- 
tremities of  the  arc. 

Comparing  the  result  here  given  with  Art.  107,  we  see 
that  the  time  of  describing  any  arc  of  a  circle  is  in  this  case 
in  a  constant  ratio  to  the  time  of  describing  a  corresponding 
arc  of  a  second  circle,  in  which  the  motion  is  oscillatory. 

The  time  of  describing  any  arc  of  the  circle  is,  in  general, 
an  elliptic  function.  There  is  one  case,  however,  in  which  it 
admits  of  a  simple  expression,  viz.,  where  DR  is  a  tangent  to 
the  circle,  as  in  Art.  107. 

In  this  case  we  have  h  -  1,  and  the  definite  integral  be- 
comes 


J/JCOS0 


The  time  of  motion  from  any  point  to  the  highest  point 
in  the  circle  becomes  infinite  in  this  case,  as  already  observed 
in  Art.  107  ;  accordingly  the  particle  would  continually  ap- 
proach the  highest  point  without  ever  reaching  it. 

115.  Pressure  on  Curve. — If  m  denote  the  mass  of  the 

particle,  then  the  normal  pressure  R  on  the  circle  consists  of 

two  parts — one  arising  from  the  centrifugal  pressure,  the  other 

from  the  weight — hence  we  get 

v2 
R  =  m  —  +  mg  cos  0 
a 


=  m^  +  3cos0 


where  CD  =  d. 


Pressure  on  Curve.  119 

At  the  lowest  point  this  becomes  mg  [  —  +  3  \  at  the 

highest  point,  mg[ 3  J;  and  when  the  string  is  horizontal, 

2d  \«        / 

mgH' 

If  the  particle,  instead  of  moving  in  a  tube,  is  attached  by 
a  string,  of  length  «,  to  a  fixed  point  C,  and  thus  constrained 
to  move  in  a  vertical  circle,  the  preceding  expression  gives 
the  tension  of  the  string  for  any  position.  As  long  as  the 
tension  is  positive  the  string  remains  stretched.  At  the  point 
where  R  =  0  the  tension  vanishes,  and  the  particle  will  leave 
the  circle  and  proceed  to  describe  a  parabola.  It  is  immediately 
seen  that  the  distance  of  this  point  from  the  line  DR  is  one- 
third  of  CD  (see  fig.  of  last  Article). 

These  results  will  be  illustrated  by  the  following  ex- 
amples : — 

Examples. 

1.  A  particle  slides  down  the  convex  side  of  a  vertical  circle ;  determine  the 
point  at  which  it  will  leave  the  curve. 

Here,  since  the  velocity  at  the  highest  point  on  the  circle  is  zero,  we  have 
d  =Jf;  accordingly  the  point  at  which  R  =  0  is  given  by  the  equation 
cos  0  =  -  f .     The  geometrical  construction  is  evident. 

2.  A  particle  is  projected  from  the  lowest  point  along  the  inside  of  a  smooth 
vertical  circle ;  find  the  least  velocity  of  projection  in  order  that  the  particle 
should  make  a  complete  revolution  in  the  circle.  Ans.  \Jbag. 

In  this  case  the  pressure  at  the  highest  point  is  zero,  and  at  every  other  point 
is  positive. 

3.  If  the  initial  velocity  he  less  than  that  in  the  preceding  example,  find 
the  point  P  at  which  the  particle  will  leave  the  circle,  and  where  it  will  strike 
it  again. 

The  construction  for  the  point  P  in  question  has  been  given  above.  After 
leaving  the  circle  the  particle  describes  a  parabola,  and  the  point  Q  in  which 
it  again  meets  the  circle  is  found  by  drawing  PQ,  making  with  the  vertical 
direction  an  angle  equal  to  that  which  the  tangent  at  P  makes  with  the  vertical. 
This  result  follows  immediately  from  the  principle  that  the  vertical  circle 
osculates  the  parabolic  trajectory  at  P. 

4.  In  the  same  case  find  the  direction  of  motion  of  the  particle  at  the  instant 
it  returns  to  the  circle. 

Ans.  tan  £  =  £  tan  a,  where  &  is  the  angle  which  the  required  direction  of 
motion  makes  with  the  vertical ;  and  a  is  the  corresponding  angle 
at  the  point  P,  where  the  particle  leaves  the  circle. 


120 


Circular  Motion. 


5.  Find  the  velocity  of  projection  from  the  lowest  point  on  the  circle,  in 
order  that  the  particle  after  leaving  the  circle  should  meet  it  again  at  its  lowest 
point.  ji„8m  | \l\\ga. 

6.  Show  that  the  solution  of  the  general  problem  of  finding  the  initial  velo- 
city, in  order  that  the  particle  after  leaving  the  circle  shall  meet  it  again  at  a 
given  point,  depends  on  the  trisection  of  an  arc. 

7.  A  material  particle  moves  in  a  circular  groove  on  a  smooth  inclined  plane  ; 
if  it  be  projected  from  its  point  of  rest  with  a  velocity  just  sufficient  to  carry  it 
to  the  highest  point  in  the  groove,  find  the  time  of  its  motion. 

116.  Lemma  on  Coaxal  Circles. — A  chord  PQ  of  a 
circle  touches  a  second  circle  at  0; 
and  PL,  QM  are  drawn  perpen- 
dicular to  the  radical  axis  of  the 
two  circles  :  to  prove  that 

P02:Q02=PL:QM. 

Let  R  be  the  point  of  intersec- 
tion of  PQ  with  the  radical  axis ; 
then,  since  the  tangents  from  R  to 
the  circles  are  of  equal  length,  we 
have  R02  =  RP.RQ; 


therefore 
Consequently 


or 


RQ:  RO=RO:RP. 
QO:  OP=RQ:RO, 
Q02:OP2=RQ2:RQ.RP 

=  RQ:RP  =  QM:PL. 


(33) 


If  now  we  suppose  a  particle  to  describe  the  outer  circle 
with  a  velocity  due  to  the  level  LM,  and  P'Q'  be  drawn 
indefinitely  near  to  PQ,  touching  the  inner  circle,  these 
tangents  may  be  regarded  as  intersecting  in  0,  and  we 
accordingly  have 

PF :  QQ'  =  PO :  QO  =  yPL :  -/  QM . 

Again,  let  v,  v  be  the  velocities  of  the  particle  when  at  P 
and  Q  respectively ;  then  v2  =  2gPL,  v2  =  2g  QM; 


therefore        v:v =</PL  :  </QM=  PP' :  Q Q'. 


Application  to  Elliptic  Functions.  121 

Hence  the  time  of  describing  PPf  is  the  same  as  that  of 
describing  QQ';  consequently  the  time  of  motion  from  P  to 
Q  is  the  same  as  that  from  P'  to  Q',  and  hence  we  readily 
infer  that  the  time  is  the  same  for  the  description  of  all  arcs 
cut  off  by  tangents  drawn  to  the  inner  circle. 

Examples. 

1.  A  particle  is  moving  in  a  smooth  vertical  circle  under  the  action  of 
gravity :  the  time  of  description  of  a  variable  arc  of  the  circle  being  supposed 
constant,  show  that  the  envelope  of  its  chord  is  another  circle. 

2.  Show  that  if  the  time  of  motion  from  Pto  Q  be  the  same  as  that  from 
the  highest  to  the  lowest  point  on  the  circle,  the  line  PQ  always  passes  through 
a  fixed  point. 

3.  Two  particles  are  projected  from  the  same  point,  in  the  same  direction, 
and  with  the  same  velocity,  but  at  different  instants,  in  a  smooth  circular  tube, 
of  small  bore,  whose  plane  is  vertical.  Prove  that  the  line  joining  them  always 
touches  a  circle. 

4.  In  the  same  case,  if  the  particles  be  projected  in  opposite  directions,  the 
other  circumstances  being  unaltered,  prove  that  the  line  joining  their  positions 
always  touches  a  circle ;  and  find  when  the  circle  becomes  a  fixed  point. 

5.  A  particle  is  moving  in  a  vertical  circle  under  the  action  of  gravity.  If 
three  points  L,  M,  iVbe  taken  on  the  circle,  find  a  fourth  point  P,  such  that  the 
time  of  motion  from  JVto  P  shall  be  equal  to  that  from  L  to  M. 

6.  In  the  same  case  find  P,  so  that  the  time  of  describing  NP  shall  be  double, 
or  any  given  multiple  of  that  of  describing  LM. 

7.  AB  is  the  vertical  diameter  of  a  fine  circular  tube  in  which  move  three 
equal  particles  P,  Q,  Q'  (modulus  of  restitution  =  1  for  any  pair)  ;  P  starts  from 
A,  and  Q,  Q',  in  opposite  senses  from  B  with  such  velocities  that  at  the  first 
impact  all  three  have  equal  velocities ;  prove  that  throughout  the  motion  the 
line  joining  any  pair  is  either  horizontal  or  passes  through  one  of  two  fixed 
points,  and  that  the  intervals  of  time  between  successive  impacts  are  all  equal. 

Camb.  Trip.,  1874. 

117.  Application  to  Elliptic  Functions. — Since  the 
time  of  description,  under  the  action  of  gravity,  of  any  arc  of  a 
vertical  circle,  is  expressible  by  a  definite  integral  of  the  form 

the  results  of  the  last  Article  have  important  applications  in 
the  theory  of  elliptic  functions.  For  example,  they  furnish 
us  with  simple  methods  for  the  addition,  subtraction,  and 
multiplication  of  such  functions,  depending  on  elementary 
properties  of  coaxal  circles.  This  connexion  was  first  pointed 
out  by  Jacobi  (Crelle's  Journal,  1828 ;  Liouville's  Journal, 
1845). 


122  Circular  Motion. 


Examples. 

1.  Prove  that  the  time  of  descending  any  small  arc  terminated  at  the  lowest 
point  of  a  vertical  circle  is  to  the  time  down  its  chord  as  ir :  4. 

2.  If  the  length  of  a  seconds  pendulum  be  39-14  inches  ;  find  the  corre- 
sponding value  of  g  to  two  places  of  decimals. 

3.  A  clock  loses  4  minutes  in  a  day  ;  find  how  much  its  pendulum  should 
be  shortened  in  order  that  it  may  keep  correct  time.         Ans.  Its  xioth  part. 

4.  Assuming  the  length  L  in  inches  of  a  seconds  pendulum  at  the  latitude  A 
to  be  given  by  the  formula 

Z  =  39-118  -i1,;  cos  2a; 

find  the  ratio  of  the  difference  between  the  values  of  polar  and  equatorial  gravity 
to  equatorial  gravity.  Ans.  19l  69-. 

5.  Find  the  correction  in  the  time  of  vibration  of  a  circular  pendulum  when 
the  amplitude  of  the  vibration  is  30°. 

6.  If  two  particles  be  connected  by  an  inextensible  string,  and  if  one  be 
made  to  move  as  if  under  the  action  of  a  constant  force  ;  prove  that  the  relative 
motion  of  the  other  is  that  of  a  simple  pendulum. 

7.  A  series  of  smooth  circles  in  a  vertical  plane  have  a  common  highest 
point ;  a  particle  starting  at  this  point  slides  down  the  convex  side  of  each  circle  ; 
find  the  locus  of  the  point  where  the  particles  leave  the  circles. 

8.  A  mass  ms  after  sliding  down  the  inner  surface  of  a  smooth  hemispherical 
bowl,  strikes  a  mass  tn  placed  at  the  lowest  point  of  the  bowl.  If  both  bodies 
be  perfectly  elastic,  find  the  heights  to  which  they  respectively  ascend  after 
collision. 

9.  If  the  length  of  a  conical  pendulum  be  1  foot,  and  the  weight  attached 

to  its  extremity  be  1  lb.  ;  find  approximately  the  tension  of  the  connecting  wire 

when  the  time  of  its  revolution  is  one  second.     Find  also  approximately  the 

angle  which  the  revolving  wire  makes  with  the  vertical  spindle. 

4tt2  g 

Ans.  Tension  =  —  lb. ;  cos  0  =  — — . 
g  4tt~ 

10.  Investigate  the  motion  of  a  cycloidal  pendulum  when  acted  on  by  a 
constant  force  /,  always  in  a  direction  opposite  to  that  of  its  motion,  in  addition 
to  the  force  of  gravity. 

d2s       g  . 

Here  the  equation  of  motion  is         -jz>  +  ;r-  *  =/> 

(it"  Ad 

and  we  get,  by  Art.  110,        *  =  —  +ccos^/^  *  +  c'sin  ^~^t. 


Examples.  123 

,  ds 

If,  when     t  =  0,     s  =  s     and     —  =  0,     we  get 

2a/      /,      2«A  |7,         ,«&         /.      2«A    (7    •       (7, 

*=— —  +  I* -    cosJ  — £,  and  -r  =-  (  s J     — -  sin      —  £. 

y         \  £  /       \2«  dt         \  g  )\2a         \2« 

This  vanishes  when  J ^  =  7r;  accordingly  the  time  of  an  oscillation  is  n  J — ; 
the  same  as  when  unresisted. 

1 1 .  A  heavy  particle  is  connected  by  an  inextensible  string,  3  feet  long, 
to  a  fixed  point,  and  describes  a  circle  in  a  vertical  plane  about  that  point, 
making  600  revolutions  per  minute  ;  find,  approximately,  the  ratios  of  the  ten- 
sions of  the  string  when  the  particle  is  at  the  highest  and  lowest  points,  and 
when  the  string  is  horizontal. 

12.  A  body  hangs  freely  from  a  fixed  point  by  an  inextensible  string  2  feet 
in  length.  It  is  projected  "in  a  horizontal  direction  with  a  velocity  of  20  feet 
per  second.  Compare  the  tensions  at  the  highest  and  lowest  points  of  the  circle 
which  is  described,  assuming  g  =  32.  -4ws.  29  :  5. 

13.  Show  that  the  time  of  a  small  oscillation  of  a  pendulum  which  vibrates 
in  the  air  is  unaffected  by  its  resistance. 

The  resistance  is  usually  assumed  to  vary  as  the  square  of  the  velocity.     It 

(dd\z 
can  accordingly  be  expressed  by  a  term  of  the  form  fx  (  —  J  ,  where  fi  is  a  very 

small  fraction ;  hence  in  this  case  the  equation  of  motion  may  be  written 


d°-9     g  ldd\  2 

dt2      I 


(dQ\ 


Since  /t  is  small,  as  also  ^,  we  get  as  a  first  approximation  0  =  o  cos  J-  t, 


dQ_ 
dt* 

'de\2 


as  before.     If  this  value  be  substituted  in  p  l  —  \  ,  in  accordance  with  the 
method  of  successive  approximations,  the  differential  equation  becomes 


de 

dt 
t  =  0,  is 


The  integral  of  this,  subject  to  the  condition  that  0  =  o,  and  ^  = 


0  =  1/xd2  +  (a  -  f  ,ua2)  cos  t. 


^+iaVccs2^ 


124  Circular  Motion. 

Also  ^  =  ~yjjsin  wf  [a " Ia"*2  +  *  *"* cos  wf )  ' 

Hence,  since  —  =  0  at  the  end  of  one  vibration,  if  T  be  the  corresponding 

value  of  t,  we  have  smTJj  =  0,  or  T=  irJ--     Accordingly,  the  duration  of 

the  oscillation  is  not  affected  by  the  resistance.     Also,  since  we  have  in  this 

case,  cos  t  A-  =  -  1,  the  corresponding  value  of  0  is  -  (a-  f  ^o2) ;  accordingly 

the  resistance  of  the  air  reduces  the  amplitude  of  the  oscillation  by  f  jxa2.  The 
successive  angles  of  oscillation  diminish  according  to  the  same  law,  but  the  time 
of  oscillation  remains  the  same  for  each. 


(    125    ) 


CHAPTER  VI. 


WORK    AND     ENERGY. 


118.  Work. — In  all  cases  where  force  is  employed  in  over- 
coming resistance  so  as  to  produce  motion,  work  is  said  to  be 
performed.  Hence  the  conception  of  work  involves  both 
motion  and  resistance ;  and  therefore  a  corresponding  effort 
or  force  to  overcome  the  resistance.  In  general,  work  may- 
be defined  as  the  act  of  producing  a  change  in  the  configu- 
ration of  a  system  in  opposition  to  forces  which  resist  that 
change.  We  proceed  to  consider  how  the  amount  of  work 
performed  in  any  case  is  to  be  estimated. 

119.  Measure  of  Work. — The  simplest  idea  of  work 
is  derived  from  raising  a  weight  through  a  vertical  height ; 
in  which  case  the  attracting  force  of  the  Earth  is  the  resistance 
overcome.  The  amount  of  work  in  such  cases  evidently  in- 
creases in  proportion  to  the  weight  of  the  body  raised  and  to 
the  height  to  which  it  is  raised.  For  example,  the  work  done 
in  raising  one  ton  through  a  height  of  10  feet  is  ten  times 
that  of  raising  it  one  foot,  or  twenty  times  that  of  raising  one 
cwt.  through  10  feet;  and  so  on  in  all  cases.  Hence  it  is 
readily  seen  that  the  work  performed  in  such  cases  is  measured 
by  the  product  of  the  weight  into  the  height,  i.e.  by  Wh,  where 
W  represents  the  number  of  units  in  the  weight,  and  h  that 
in  the  height. 

In  general,  if  we  confine  our  attention  to  a  single  point 
which  is  moved  in  direct  opposition  to  a  constant  resisting 
force,  the  work  done  is  estimated  by  the  product  of  the  force 
and  the  distance  through  which  the  point  is  moved,  i.e.  by  Ppf 
where  P  represents  the  force,  which  overcomes  the  equal  and 
opposite  resisting  force,  and  p  the  distance  passed  over. 

120.  Gravitation  Unit  of  Work. — From  the  ordi- 
nary units  adopted  in  this  country  we  derive  the  unit  of 
work  called  a  footpound,  i.e.  the  work  performed  in  raising 


126  Work  and  Energy. 

one  pound  through  one  foot  in  height.  This  is  the  unit 
usually  adopted  in  practical  local  application  of  work,  and  is 
called  the  Gravitation  Unit  of  Work  (Art.  65).  The  corre- 
sponding unit  in  the  metric  system  is  called  the  kilogram- 
metre,  or  kgm.  That  is  the  work  of  raising  a  kilogramme 
through  the  height  of  a  metre.  A  kilogrammetre  is  7*233 
foot-pounds.  The  unit  of  work  in  this  system  varies  slightly 
from  place  to  place  with  the  value  of  g,  and  this  should  be 
remembered  if  numerical  or  scientific  accuracy  were  required 
(Art.  39). 

121.  Absolute  Unit  of  Work. — In  the  absolute  sys- 
tem the  unit  of  resistance  is  that  already  adopted  (Art.  64) 
as  the  unit  of  force.  Thus,  if  we  take  a  poundal  as  the  unit 
of  force,  the  corresponding  unit  of  work  is  that  done  by  a 
poundal  acting  through  a  foot.  This  is  sometimes  called  the 
foot-poundal.  It  is  obvious  that  a  foot-pound  is  g  times  a 
foot-poundal :  accordingly,  any  result  in  the  former  system  is 
reducible  to  the  latter  at  any  place  by  multiplying  by  the 
corresponding  value  of  g. 

Again,  adopting  the  definition  of  a  dyne  given  in  Art.  64, 
the  work  done  by  a  dyne  in  working  through  a  centimetre gis 
called  an  erg ;  and  a  foot-poundal  is  421,394  ergs. 

In  such  measurements  as  are  required  in  electrical  and 
magnetic  investigations,  the  absolute  unit  of  work  is  always 
adopted,  and  the  erg  is  the  unit  usually  employed. 

122.  Horse-power. — Although  in  our  definition  of 
work  we  have  taken  no  account  of  the  time  occupied  in  its 
performance,  yet  time  becomes  a  necessary  element  when  we 
come  to  compare  the  efficiency  of  different  agents.  For  in- 
stance, if  one  agent  working  uniformly  performs  an  amount 
of  work  in  one  hour  which  it  requires  another  5  hours  to 
accomplish,  the  former  is  said  to  be  five  times  as  efficient. 
In  comparing  the  work  done  by  a  steam-engine  or  other 
agent  we  usually  adopt  as  our  unit  the  horse-power  defined 
by  Watt. 

Thus  an  engine  is  said  to  be  of  one-horse-power  when  it 
is  capable  of  performing  33,000  foot-pounds  of  work  in  one 
minute  of  time,  or  550  foot-pounds  in  one  second,  and  so  on 
in  proportion. 


Horse-power.  127 

Continental  writers  employ  horse-power  as  75  kgm.,  that 
is,  542*475  foot-pounds,  per  second. 

123.  Again,  the  work  performed  in  raising  a  body  of 
weight  W  to  any  height  h  is  the  same  whether  the  body 
be  raised  vertically  up  or  brought  up  by  any  other  course. 
The  whole  work  is  still  represented  by  Wh,  where  h  is  the  space 
through  which  the  weight  has  been  moved,  estimated  in  the 
vertical  direction,  i.  e.  in  that  in  which  the  resistance  of 
gravity  acts.  And,  generally,  the  work  done  by  any  uniform 
effort  or  force,  acting  in  a  constant  direction  against  an  equal 
and  opposite  force  P,  is  measured  by  the  product  of  the  force 
into  the  space  through  which  its  point  of  application  is  moved, 
estimated  in  the  direction  in  which  the  force  acts. 

Thus,  if  a  force  P  be  supposed  to  act  at  A,  and  to  move  its 
point  of  application  to  B ;  then  if  BM  be 
drawn  perpendicular   to   AP,   the   work 
done  is  estimated  by  Pp,  or  by  PAs .  cos  0, 


where  p  =  AM,  As  =  AB,  and  0  =  L  BAM.  A     M 

The  work  done  is,  therefore,  regarded  as  positive  or 
negative  according  as  the  angle  0,  which  the  direction  of  the 
force  makes  with  that  of  the  motion,  is  acute  or  obtuse. 

If  6  =  \tt,  the  direction  of  the  motion  is  perpendicular  to 
that  of  the  force,  and  the  work  done  is  zero. 

If  two  or  more  forces  act  on  a  system,  the  whole  work 
done  is  the  sum  of  the  works  done  by  each  force  separately. 

If  any  number  of  forces  be  in  equilibrium,  it  can  be  readily 
seen  that  the  total  work  done  by  them  for  any  small  dis- 
placement is  zero  :  from  this  the  statical  principle  of  virtual 
velocities  can  be  immediately  deduced. 

Examples. 

1.  Prove  that  the  -whole  work  done  in  raising  a  system  of  heavy  bodies,  each 
through  a  different  height,  is  the  same  as  that  of  raising  their  entire  weight 
through  a  height  equal  to  that  through  which  their  centre  of  inertia  is  raised. 

2.  Find  the  work  performed  in  moving  a  ton  along  100  yards  on  a  uniformly 
rough  horizontal  road,  the  coefficient  of  friction  being  -rV. 

Ans.  67,200  foot-pounds. 

3.  Show  that  the  same  work  is  expended  in  drawing  a  body  up  an  inclined 
plane,  subject  to  friction,  as  would  be  expended  upon  drawing  it  first  along  the 
base  of  the  plane  (supposing  the  coefficient  of  friction  the  same),  and  then  raising 
it  up  the  height  of  the  plane. 


128  Work  and  Energy. 

4.  "What  time  will  10  men  take  to  pump  the  hold  of  a  ship  which  contains 
30,000  cubic  feet  of  water;  the  centre  of  inertia  of  the  water  being  14  feet 
below  the  point  of  discharge,  and  each  man  being  supposed  to  perform  1500  foot- 
pounds per  minute  ;  assuming  the  weight  of  a  cubic  foot  of  water  to  be  62^  lbs.  ? 

Ans.  29  hrs.  10  mins. 

124.  Work  done  by  a  Variable  Force. — If  the  force 
be  not  constant,  we  may  suppose  the  path  described  by  its 
point  of  application  divided  into  portions  so  small  that  for 
each  the  force  may  be  considered  constant.  Hence,  for  the 
displacement  ds  of  its  point  of  application,  Pds  is  the  corre- 
sponding element  ofivorks  and  the  total  work  in  moving  through 

any  space  s  is  represented  by  the  definite  integral     Pds. 

J  0 

If  the  direction  of  P  makes  an  angle  0  with  ds,  the  cor- 
responding element  of  work  is  P  cos  6ds,  and  the  total  work 
is  represented  by 

Pcos  Qds. 

Again,  let  x,  y,  z,  be  at  any  instant  the  coordinates  of  the 
point  of  application  of  the  force  P,  referred  to  a  system  of 
rectangular  axes ;  and  let  X,  Y,  Zy  be  the  components  of  P 
parallel  to  the  coordinate  axes  respectively  ;  then  we  have 

Pcos  0  ds  =  Xdx  +  Ydy  +  Zdz. 

Hence  the  total  work  done  by  P  in  moving  its  point  of 
application  from  one  point  to  another  is  represented  by 

{Xdx  +  Ydy  +  Zdz) 

taken  between  the  two  points. 

If  the  expression  Xdx  +  Ydy  +  Zdz  be  an  exact  diffe- 
rential, i.  e.  if 

Xdx  +  Ydy  +  Zdz  =  du9 

where  u  is  a  function  of  x,  y,  z,  then  the  integral 

\(Xdx  +  Ydy+  Zdz), 

taken  between  any  two  points,  is  a  function  of  the  coordinates 
of  those  points  ;  and  the  work  done  is  accordingly  a  function 


Forces  directed  to  Fixed  Centres.  129 

of  the  extreme  coordinates  solely.  When  this  is  so,  the 
mutual  forces  between  the  parts  of  a  system  always  perform 
or  always  consume  the  same  amount  of  work  during  any 
motion  whatever  by  which  it  can  pass  from  the  one  particular 
configuration  to  the  other ;  hence  such  a  system  is  called  a 
conservative  system  of  forces.  In  general,  for  any  system  of 
forces  acting  at  different  points,  the  total  work  W  done  for 
any  finite  displacements  is  represented  by 


TT=S 


Pdp  =  2 


{Xdx  +  Ydy  +  Zdz),  (1) 


where  the  summation  extends  to  all  the  forces  of  the  system. 
125.  Forces  directed  to  Fixed  Centres.    Potential. 

— If  the  force  F  be  directed  to  a  fixed  centre,  and  if  r  be  the 
distance  of  its  point  of  application  from  the  centre,  then  the 
corresponding  element  of  work  is  represented  by  Fdr;  and 
the  total  work,  when  the  point  is  moved  from  a  distance  /  to 

a  distance  r",  is  represented  by      Fdr. 

Jr 

If  F  be  a  function  of  r  represented  by  n<j>'(r),  then  the 
value  of  this  integral  will  be 

m!*(0-  *('•'))• 

In  the  law  of  attraction  which  holds  in  nature  we  have 
F=-  —  ;  and  the  expression  /u(  — ,  J  represents  the  corre- 
sponding work  in  moving  a  unit  of  mass  from  the  distance 
/  to  the  distance  /'.  Hence  the  work  done  in  the  motion 
of  a  unit  mass  from  an  infinite  distance  to  the  distance  r  is 

represented  by  -. 

The  function  2  —  in  the  case  of  the  ordinary  law  of  gravi- 
tation is  called  the  potential  of  the  system  of  attracting 
masses.  This  potential  function  is  usually  represented  by 
V ;  and  if  dm  be  the  element  of  attracting  mass,  and  r  its 
distance  from  a  point  P,  then  V,  the  potential  at  P,  is 
denoted  by  dm 

V=*-  (2) 

extended  through  all  points  in  the  attracting  system. 

K 


130  Work  and  Energy. 

Again,  if  a  number  of  forces  F,  F\  F'\  &c,  be  directed 
to  fixed  centres,  and  if  r,  /,  r",  &c,  be  the  corresponding 
distances,  then  the  total  work  is  represented  by 

\Fdr  +  \F'drf  +  \F"dr"  +  &c, 
taken  between  the  limiting  positions. 

If  the  forces  be  each  a  known  function  of  the  distance 
from  the  corresponding  centre  of  force,  the  result  can,  in  ge- 
neral, be  immediately  integrated,  and  the  work  is  a  function 
of  the  initial  and  final  positions  of  the  points  of  application 
solely.  Consequently  such  a  system  of  forces  is  always  a 
conservative  system. 

Example. 
If  m.  m  be  the  masses  of  two  particles  attracting  each  other  with  a  force 

u*^-  where  r  is  their  distance  apart,  show  that  the  work  done  when  they  have 
r2  ' 

mm 
moved  from  an  infinite  distance  apart  to  the  distance  ris  /x  ■ -. 

126.  Potential  of  an  Attracting  Spherical  Mass.— 

If  each  element  of  the  surface  of  a  sphere  be  divided  by  its 
distance  from  an  external  point,  and  the  sum  taken  over  the 
entire  surface,  this  sum  is  readily  shown  by  elementary 
integration  to  be  equal  to 

S 

~$ 

where  S  is  the  whole  surface  of  the  sphere,  and  R  the  distance 
from  its  centre  to  the  external  point. 

Hence,  if  a  mass  m  be  uniformly  spread  over  the  surface 
of  the  sphere  8,  we  have 

a*.   » 

r       Jx 

From  this  it  follows  at  once  that  in  a  solid  sphere  of  mass 
M,  for  which  the  density  is  constant  through  each  concentric 
couche,  we  have 

r-x£-J  (4) 

r       Jx 

That  is,  the  potential  is  the  same  as  if  the  whole  mass 
were  concentrated  at  the  centre  of  the  sphere. 

Consequently  the  work  done  by  an  attracting  sphere  M, 


Work  done  by  a  Stress.  131 

in  moving  a  unit  of  mass  from  the  distance  R'  to  the  distance 
jR,  measured  from  the  centre,  is 

It  may  be  remarked  that  it  can  be  readily  seen  from  (4) 
that  a  homogeneous  sphere  attracts  an  external  mass  as  if 
the  whole  mass  of  the  sphere  were  concentrated  at  its  centre. 

127.  Work  done  by  a  Stress. — If  two  equal  and  oppo- 
site forces,  each  represented 

by  F,  act  respectively  at  the  Ar  B 

points  A  and  B,  along  the  f\~~  A 

line  connecting  these  points,     p      M  A  B    N       F 

to  find  the  element  of  work 

for  a  small  displacement.  Suppose  A!  and  B  to  be  the  new 
positions  for  an  indefinitely  small  displacement,  and  let  fall 
the  perpendiculars  AM  and  B'N  on  the  line  AB ;  then  the 
elements  of  work  are  represented  by  F.  AM  and  F.  BN. 
Hence  their  sum  is  F{AM  +  BN)  =  F(AB'  -  AB),  or  FAs, 
where  As  denotes  the  indefinitely  small  change  in  the  distance 
between  the  points  of  application  of  the  forces. 

Hence,  if  the  points  A  and  B  be  rigidly  connected,  as  the 
distance  AB  is  invariable,  the  total  work  done  by  the  forces 
for  any  displacement  is  zero. 

Also  the  point  of  application  of  a  force  maybe  transferred 
from  any  one  point  to  any  other  on  its  line  of  action  without 
altering  the  work  done,  provided  the  distance  between  the 
two  points  is  invariable. 

The  pair  of  equal  and  opposite  forces  that  two  bodies 
exert  on  one  another  in  accordance  with  the  general  prin- 
ciple of  action  and  reaction  is  called  in  modern  treatises  a 
stress.  When  the  forces  act  away  from  each  other,  as  in  the 
figure,  the  stress  is  called  a  tension ;  when  they  act  towards 
each  other  it  is  called  a  pressure. 

Hence  the  work  done  by  a  stress  is  positive  or  negative 
according  as  the  change  of  distance  between  the  points  of 
application  is  in  the  direction  of  the  mutual  action  of  the 
forces  or  in  the  opposite  direction. 

Also  in  the  case  of  a  rigid  body  it  follows  that  the  total 
work  done  by  the  internal  forces  of  stress  is  always  zero, 

k2 


132  Work  and  Energy. 

128.  Body  with  a  Fixed  Axis.— To  find  the  work 
done  by  a  force  acting  on  a  rigid  body  which  is  capable  of 
turning  round  a  fixed  axis. 

Suppose  the  force  R  resolved  into  two  components — one 
parallel,  the  other  perpendicular  to  the 
fixed  axis.     The  former  does  no  work, 
since  it  is  perpendicular  to  the  direction 
of  motion  of  every  point  in  the  body. 

Let  the  latter  component  be  repre- 
sented by  P,  and  suppose  it  to  act  in  the 
plane  of  the  paper ;  the  fixed  axis  being 
perpendicular  to  that  plane,  and  meeting  it  in  the  point  0. 
Let  N  be  the  foot  of  the  perpendicular  drawn  from  0  to  the 
line  of  action  of  P ;  then  by  the  last  Article  we  may  take  i^as 
the  point  of  application  of  P. 

Suppose  now  the  body  to  receive  a  small  angular  displace- 
ment Ad  round  the  fixed  axis  in  the  direction  of  the  arrow ; 
then,  if  ON '  =  p,  the  displacement  of  N  will  be  p&6,  and  the 
corresponding  element  of  work  is  P/;A0,  or  A0  multiplied 
by  the  moment  of  the  force  R  with  respect  to  the  fixed  axis. 

Again,  if  we  suppose  a  pair  of  equal,  parallel,  and  opposite 
forces  to  act  on  the  rigid  body  ;  then,  provided  the  plane  of 
the  pair  is  perpendicular  to  the  fixed  axis,  the  work  clone  by 
the  pair  is  evidently,  from  what  precedes,  represented  by  the 
moment  of  the  pair  multiplied  by  the  small  angle  of  rotation. 
And  if  the  pair  continue  to  act  on  the  body,  the  work  done  by 
it  during  any  rotation  is  represented  by  the  product  of  the 
moment  of  the  pair  by  the  angle,  in  circular  measure,  through 
which  the  body  has  rotated. 

Example. 

A  pivot  or  screw  turns  round  a  central  axis  and  presses  against  a  rough, 
plane  ;  find  an  expression  for  the  work  expended  on  the  friction  which  acts  on 
the  circular  end  of  the  pivot  in  one  revolution  round  its  axis. 

Let  Q  denote  the  entire  normal  pressure  between  the  pivot  and  the  plane, 
/j.  the  coefficient  of  friction,  supposed  constant,  a  the  radius  of  the  end  of  the 
pivot.  This  end  may  he  regarded  as  consisting  of  an  indefinitely  great  number 
of  concentric  circular  rings.  If  r  he  the  radius  of  one  of  the  rings,  dr ^  its 
breadth,  then  the  area  of  the  ring  is  27rrdr,  and  the  corresponding  friction, 

taken  over  the  entire  ring,  is  represented  by — ~rdr.   Hence  the  corresponding 


Measure  of  Kinetic  Energy.  133 

-work  for  one  revolution  is  — —r'2dr.    Integrating,  we  get  %ir/j.Qa  for  the  en- 
re-  f 

tire  work  expended.     In  this  investigation  the  nonnal  pressure  Q  has  been  sup- 
posed to  be  uniformly  distributed  over  the  end  of  the  pivot. 

129.  Energy. — Energy  is  the  capacity  of  doing  work. 
For  instance,  a  spring  when  bent  by  pressure  contains  a  cer- 
tain amount  of  energy  stored  up  in  it ;  thus  the  mainspring 
of  a  watch,  by  the  energy  which  it  possesses,  maintains  the 
motions  of  the  works  until  that  energy  has  been  expended. 
Again,  a  quantity  of  air,  when  compressed  into  a  smaller 
volume,  possesses  energy,  and  can  perform  work  when 
occasion  requires;  for  example,  in  projecting  a  bullet  from 
an  air-gun.  Also  a  raised  weight  is  capable  of  doing  work, 
and  is  therefore  said  to  possess  energy.  For  instance,  the 
motion  of  a  clock  is  maintained  by  the  energy  of  its  descend- 
ing weights.  The  energy  of  a  weight  IF  raised  to  a  heights 
above  the  ground  is  measured  by  Wh,  that  is,  by  the  work  it 
is  capable  of  performing  by  its  descent  to  the  ground.  In 
general,  when  the  configuration  of  a  system  is  altered,  it  has 
a  tendency  to  return  to  its  former  state,  and  in  effecting  this 
return  is  capable  of  doing  a  certain  amount  of  work.  This 
capacity  of  doing  work,  arising  from  change  of  configuration 
or  of  relative  position  in  a  system,  is  called  potential  energy  ; 
the  work  employed  in  producing  this  change  being  in  a  sense 
accumulated.  For  example,  if  two  bodies  which  attract  one 
another  are  separated,  they  have  a  tendency  to  rush  together, 
and  in  so  doing  are  capable  of  overcoming  a  certain  amount 
of  resistance. 

Again,  a  body  in  motion  possesses  a  certain  amount  of 
energy  which  is  measured  by  the  work  it  is  capable  of  per- 
forming before  being  brought  to  rest.  This  latter  is  called 
the  Kinetic  energy  of  the  body.  We  proceed  to  consider  how 
its  amount  is  measured. 

130.  Measure  of  Kinetic  Energy. — The  measure  of 
the  kinetic  energy  of  the  mass  m  moving,  without  rotation, 
with  the  velocity  v,  is  easily  found.  For,  suppose  the  mass 
acted  on  by  a  uniform  resistance  R  in  the  direction  of  its 
motion,  and  let  R  =  mf;  then,  if  v  be  the  initial  velocity 
and  s  the  space  described  before  coming  to  rest,  we  have,  by 
Art.  37,  v2  =  2/8 ;  hence  \mv2  =  Rs. 


J 


134  Work  and  Energy. 

Accordingly,  the  work  which  a  mass  m  moving  with  the 
velocity  v  is  capable  of  performing  before  being  brought  to 
rest  is  J  mv2.  Hence  its  kinetic  energy  is  equal  to  half  its  vis 
viva,  and  is  represented  by  \  mv2. 

Examples. 

1.  A  train  of  60  tons,  moving  at  the  rate  of  15  miles  an  hour  on  a  horizontal 
railway,  runs,  when  the  steam  is  shut  off  and  the  breaks  applied,  through  a 
quarter  of  a  mile  before  stopping.  Find  in  lbs.  the  mean  resistance,  and  its 
time  of  action.  Ans.    770  lbs.;  2  minutes. 

2.  The  breadth  of  a  river  at  a  certain  place  is  100  yards,  its  mean  depth  is 
8  feet,  and  its  mean  velocity  3  miles  an  hour.  Calculate  its  horse-power,  as- 
suming a  cubic  foot  of  water  to  weigh  62|  lbs. 

Here  the  quantity  of  water  which  passes  per  minute  is  633,600  cubic  feet ; 
and  the  required  answer  is  easily  seen  to  be  363  horse-power. 

3.  A  shot  of  1000  lbs.,  moving  at  1600  feet  per  second,  strikes  a  fixed  target. 
How  far  will  the  shot  penetrate,  the  target  exerting  on  it  an  average  pressure 
equal  to  the  weight  of  12,000  tons?  Ans.  \\  ft.,  approximately. 

4.  Determine  in  ergs  the  kinetic  energy  of  a  mass  of  one  hundred  pounds 
moving  with  a  velocity  of  one  foot  per  minute.  Ans.  5853. 

5.  A  heavy  particle  resting  on  a  rough  inclined  plane,  and  attached  by  a 
string  to  a  fixed  point  on  the  plane,  is  projected  from  the  lowest  point  of  the 
circle  in  which  it  moves  in  the  direction  of  the  tangent,  (a)  Find  the  velocity 
necessary  to  carry  the  string  to  a  horizontal  position ;  (b)  If  the  particle 
descending  from  this  position  reach  the  lowest  point  and  remain  there,  deter- 
mine the  coefficient  of  friction. 

6.  A  ball  moving  with  a  velocity  of  1000  feet  per  second  has  its  velocity 
f- reduced  by  100  feet  in  passing  through  a  plank.     Through  how  many  such 

y  ■        planks  would  it  pass  before  being  stopped ;  assuming  the  same  amount  of  work 
to  be  performed  in  overcoming  the  resistance  of  each  plank  ?  Ans.  5^. 

131.  Energy  due  to  a  Variable  Foree. — If  a  va- 
riable force  F  act  at  the  centre  of  inertia  of  a  mass  m,  in 
the  direction  of  its  motion,  we  have,  by  Art.  68, 

_.         dv  dv 

F  =  m  —  =  mv—, 
dt  ds 

or  Fds  =  mvdv; 

accordingly,  if  V0  and  V\  be  the  initial  and  final  velocities 
of  m,  we  have 


i»  ( Pi* -F.V  ('*'*•  (6) 

Jo 


Energy  due  to  a  Variable  Force.  135 

From  this  we  infer  that  if  a  variable  force  F  act  on  a 
mass  0t,  in  the  direction  of  its  motion,  the  work  done  by  it  is 
measured  by  half  the  corresponding  change  in  the  vis  viva  of 
the  moving  body,  or  by  the  change  in  its  kinetic  energy. 

In  general,  let  X,  F,  Z,  as  before,  denote  the  components, 
parallel  to  the  axes  of  x,  y,  z,  of  the  force  acting  on  the  mass 
m  ;  then,  by  Art.  68,  we  have 

_         d2x      _         d2y       „         d2z 
X=m—9     Y=m-ff      Z=m—. 
dt%  dtf  dt- 

Multiply  the  first  by  dx,  the  second  by  dy,  and  the  third 
by  dz,  and  add  ;  then 

Xdx  +  Ydy  +  Zdz  =  mi-jdx  +  -~  dy  +  -^  dz  J 

sHIHS)') 

=  \  md  (v2) . 
Hence,  if  VQ  and  Vi  be  the  initial  and  final  velocities, 

im{V?  -  V2)  -\{Xdx  +  Ydy  +  Zdz),  (7) 

the  integral  being  taken  from  the  initial  to  the  final  position 
of  the  centre  of  inertia  of  m.  Hence  we  infer  that  in  this  case 
also  the  work  done  by  the  forces  during  any  motion  is  measured 
by  half  the  change  in  the  kinetic  energy  of  the  moving  mass. 

If  after  the  lapse  of  any  time  the  velocity  of  m  become 
equal  to  its  original  value,  the  work  done  in  that  interval  by 
the  forces  which  accelerate  the  motion  is  equal  to  that  done 
by  the  forces  which  retard  it. 

In  the  case  of  a  central  force,  represented,  as  in  Art.  125, 
by  fj.<p'(r),  we  readily  obtain  the  equation 

im(v2-v'2)=v{<l>(r)-<p(r')},  (8) 

where  v'  denotes  the  velocity  at  the  distance  /  from  the  centre 
of  force.     For  the  law  of  nature,  this  becomes 

±m{v2-v'2)=fx(V-  F'),  (9) 

where  V  and  V  are  the  potentials  of  the  attraction  at  the 
distances  r  and  /,  respectively. 


136  Work  and  Energy, 

Again,  in  any  conservative  system  of  forces  the  change  in 
the  kinetic  energy  of  the  motion  under  the  action  of  the  forces, 
from  any  one  point  to  any  other,  is  a  function  of  the  coordi- 
nates of  the  points,  and  is  independent  of  the  path  described. 

132.  Equation  of  Energy. — In  general,  if  we  suppose 
any  free  rigid  body  acted  on  by  external  forces,  then  the 
total  work  done  by  the  external  forces  during  any  time  is 
equal  to  the  corresponding  change  of  the  kinetic  energy  of 
the  body. 

For  each  particle  of  the  body  moves  in  the  same  manner 
as  if  it  were  free  and  acted  on  by  forces  equal  to  those 
which  result  from  its  connexion  with  the  other  particles. 
Hence,  by  what  precedes,  the  change  in  the  kinetic  energy 
of  the  particle  is  equal  to  the  work  done  on  it  by  the  external 
forces,  together  with  the  work  due  to  the  stresses  which  arise 
from  the  action  of  the  other  particles  of  the  body  on  it. 
Accordingly,  the  total  change  in  the  kinetic  energy  of  the 
rigid  body  in  any  time  is  measured  by  the  work  done  by  the 
external  forces  in  that  time ;  since,  by  Art.  127,  the  internal 
stresses  in  this  case  do  no  work,  and  equation  (7)  may  be 
written  in  the  generalized  form 

i  S*ra  (v2  -  v02)  =  2  (Xdx  +  Ydy  +  Zdz),  (10) 

taken  between  proper  limits,  in  which  the  sign  of  summation, 
2,  is  extended  to  each  element  in  the  body. 

Examples. 

1.  A  locomotive  of  10  tons,  setting  out  from  rest,  acquires  a  velocity  of 
20  miles  an  hour  on  a  horizontal  railway,  after  running  through  a  mile  under 
the  action  of  a  constant  pressure.  Calculate  in  pounds  the  difference  between 
the  moving  and  retarding  forces,  approximately.  Arts.  57. 

2.  A  501b.  ball,  after  traversing  the  barrel  of  a  gun  of  5  feet  length,  leaves 
it  with  a  velocity  of  500  feet  per  second.  Find  approximately  the  difference 
between  the  mean  explosive  force  of  the  powder  and  mean  resistance  which  acts 
on  it.  Ans.  39062-5  lbs. 

3.  A  uniform  block  of  given  dimensions  stands,  with  one  face  perpendicular 
to  the  direction  of  motion,  on  a  railway  truck,  which  is  suddenly  stopped.  If 
the  block  be  prevented  sliding  upon  the  truck,  determine  the  speed  of  the  train 
so  that  the  block  shall  be  just  overturned. 

Here  the  kinetic  energy  of  the  block  is  expended  in  raising  its  centre  of 
gravity  until  it  is  vertically  over  the  edge  round  which  the  block  turns.  Accord- 
ingly, if  a  be  the  height  of  the  block,  and  b  the  length  of  its  edge  which  lies  in 
the  direction  of  motion,  the  required  velocity  v  is  given  by  the  equation 

V2  =  g  (^/V  +  i%  _  aj. 


Energy  of  Rotation.  137 

4.  A  catapult  is  formed  by  fixing  the  ends  of  an  elastic  string  (natural  length 
21)  at  points  A  and  A ',  at  a  short  distance  apart  on  a  horizontal  plane.  A  bullet 
placed  at  the  middle  point  of  the  string  is  drawn  back  at  right  angles  to  AA' 
(stretched  length  =  21'),  and  let  go  when  the  string  is  on  the  point  of  breaking. 
Prove  that  the  velocity  V  of  the  bullet  when  it  leaves  the  string  is  independent 
of  the  distance  A  A',  and  is  to  the  velocity  V  it  would  have  acquired  in  falling 
through  the  vertical  space  V  —  I  in  the  sub-duplicate  ratio  of  the  greatest  strain 
W  the  string  can  bear  to  the  weight  W  of  the  bullet.  Mr.  Whitworth,  Educ. 
Times. 

Let  v  be  the  velocity  of  the  bullet,  and  1r  the  length  of  the  string  at  any 
instant  during  the  motion;  then  adopting  Hooke's  law,  that  the  tension  of 
the  string  varies  directly  as  its  extension,  the  equation  of  work  becomes 


W  f1'   W 

—  r-  =  4       Jl—(r-X)dr=2W{V-l), 
u  j  i  I  —  I 


or  WV2  =  2g  {V  -V)W'=W  V'2. 

5.  The  following  extension  of  the  last  question  is  given  by  Mr.  Townsend. 
If  in  place  of  a  single  cord  there  be  n  uniform  cords,  of  the  common  unextended 
length  21,  attached  to  as  many  pairs  of  diametrically  opposite  points  on  the  cir- 
cumference of  a  fixed  circle,  and  all  drawing  the  bullet  along  the  axis  of  the  cone 
of  which  the  circle  is  the  base,  and  the  bullet  at  the  vertex ;  then  we  shall  have 

WV2  =  2ng  W  [V  -  I)  =  W  V'2, 

where  V  is  the  velocity  due  to  the  height  n  (I'  —  I). 

133.  Energy  of  Rotation. — To  find  the  kinetic  energy 
of  a  rigid  body  revolving  round  a  fixed  axis  with  an  angular 
velocity  w. 

Let  p  be  the  distance  from  the  fixed  axis  of  any  element 
dm  of  the  body ;  then  pio  will  be  the  velocity  of  dm,  and 
accordingly  the  entire  vis  viva  of  the  body 

^v-dm  =  to2  If  dm  =  w2J,  (11) 

where  I  represents  the  moment  of  inertia  of  the  rigid  body 
relative  to  the  fixed  axis  (Int.  Calc,  Art.  196).  Thus  the 
kinetic  energy  required  is  J/aA 

Examples. 

1.  The  rim  of  a  fly-wheel,  sp.  gr.  7*25,  performing  6  revolutions  per 
minute,  is  6  inches  thick,  and  its  inner  and  outer  radii  are  4  and  5  feet  respec- 
tively; calculate  its  kinetie  energy  in  foot-pounds. 

Here  a>  =  -,  and  M,  the  mass  of  the  fly-wheel  =  7 '25  x  f  x  62^  .  tt  .  lbs. 
5 
Also  (Int.  Calc,  Art.  201),  i"=  M  (Q) ;  hence  the  required  answer  is  805  foot- 
pounds, approximately. 


138  Work  and  Energy. 

2.  A  rod  of  uniform  density  can  turn  freely  round  one  end ;  it  is  let  fall 
from  a  horizontal  position ;  find  its  angular  velocity  when  it  is  passing  through 

the  vertical  position.  Am.      /_£,  where  a  is  the  length  of  the  rod. 

3.  Two  masses  if  and  31'  are  connected  as  in  Atwood's  machine  (Art.  78)  ; 
find  the  acceleration  when  the  mass  fi  of  the  revolving  pulley  is  taken  into 
account.  If  v  be  the  common  velocity  of  31  and  31'  at  any  instant,  and  fik2  the 
moment  of  inertia  of  the  pulley;  then  the  entire  vis  viva  of  the  system  is  repre- 
sented by  (M  +  31')  v2  +  fxk-  u>2. 

Hence,  if  z  be  the  distance  fallen  through  from  rest,  we  have 

(M  +  31 ' )  v2  +  (i&a2  =  2g  {31  -  M')  z. 
Also  v  =  aw ; 

.'.  v2  {{M  +  3T)  a2  +  fik2}  =  2ga2  (M -  31')  z. 

Again,  the  acceleration 

therefore 

If  the  pulley  be  supposed  a  homogeneous  cylinder,  k2  =  ^—,  and /becomes 


/= 
/= 

dv 
"di~ 
ga'< 

dv 
az 
{M- 

31') 

{31  + 

31')  a 

2  +  fd? 

M+M'+fr 

4.  Find  in  the  same  case  the  tensions  of  the  strings. 

v       2M'ai  +  lxk2  ^         23fa2  +  nk2 

AnS'     M9{M+M')a^^     Mg{M  +  W)a2+^ 

For  a  homogeneous  pulley  these  become 

__           W+fl  i3f+ ix 

Mg  — ,     and    M.  g 


2{3f+3f')  +  fJ.'  "2{M  +  M')  +  fi 

5.  A  homogeneous  cylinder,  of  weight  IF,  is  rotating  round  its  axis,  sup- 
posed horizontal,  with  an  angular  velocity  w  ;  find  to  what  height  it  is  capable 
of  raising  a  given  weight  F,  before  coming  to  rest. 

r2<a2  W 

Ans.   — ,  where  r  is  the  radius  of  the  cylinder. 

4g     P 

134.  Vis  Viva  of  any  System. — If  x,  y,  i  be  the  co- 
ordinates of  the  centre  of  gravity  of  any  moving  system  of 
masses  at  any  instant,  x,  y,  z  the  coordinates  of  the  element 
dm  at  the  same  instant ;  also,  if  £,  rj,  £  be  the  coordinates 
of  dm  relative  to  a  system  of  parallel  axes  drawn  through  the 
centre  of  gravity;  then,  as  in  Art.  14,  we  have,  adopting 
Newton's  notation, 

x  =  x  +  £,    y=y  +  v,    z  =  z  +  t; 


Vis  Viva  of  any  System.  139 

consequently, 

2v~dm  =  Srf/n  { {i  +  tf  +  (y  +  ii)2  +  (i  +  t)~) . 

Again,  if  F  be  the  velocity  of  the  centre  of  gravity,  and 
v  the  velocity  of  dm  relative  to  the  centre  of  gravity,  we  have 

V2  =  (*)2  +  {iff  +  (I)8,    v2  =  t  +  v2  +  t ; 
also 

Hence  we  get 

Sr  dto  =  F22  efcw  +  2t/2  tftw.  (12) 

Accordingly,  the  vis  viva  of  the  system  at  any  instant  con- 
sists of  two  parts,  one  of  which  is  the  vis  viva  of  the  entire  mass 
supposed  concentrated  at  the  centre  of  gravity  ;  the  other  is 
the  vis  viva  of  the  system  relative  to  the  centre  of  gravity 
regarded  as  a  fixed  point.  This  result  obviously  holds  good 
also  for  the  kinetic  energy  of  the  motion. 

Examples. 

1.  A  homogeneous  cylinder  rolls,  without  slipping,  down  a  rough  inclined 
plane,  under  the  action  of  gravity ;  investigate  the  motion. 

Since  the  motion  is  one  of  pure  rolling,  the  line  of  contact  of  the  cylinder 
and  plane  at  any  instant  may  be  regarded  as  fixed  ;  accordingly  the  friction 
acting  along  the  plane  does  no  work.  Also,  by  Art.  133,  the  kinetic  energy  at 
any  instant  is  represented  by  \ur  I,  where  i"  is  the  moment  of  inertia  of  the 
cylinder  with  respect  to  the  edge  in  contact  with  the  plane.  But  /=  M  (a2  +  k~), 
where  a  is  the  radius  of  the  cylinder,  and  Mb2  its  moment  of  inertia  relative  to 
the  axis  through  its  centre.     Hence  the  equation  of  work  gives 

Mar  (a-  +  k2)  =  2g  Ms  sin  i, 
where  5  is  the  space  down  the  plane  described  from  rest.     Consequently, 

2gs  sin  i        ..       as 

ur  =  — rr-  •     Also,  —  =  v  =  ao> ; 

a-  +  kz  at 

.     '     2gais  sin 
hence  (  —  )  =  — „  ,   ,  3 


(ds\  3     2gdls  si 
It)  =     a2  +  k 

therefore,  by  differentiation, 


d-i      a2g  sin  i 

a- 1       a-  +  k2 


140  Work  and  Energy. 

This  shows  that  the  acceleration  down  the  plane  is  constant.  Hence  the 
velocity  acquired,  and  the  space  described  in  any  time,  can  at  once  be  determined. 
If  the  cylinder  be  homogeneous,  we  have  k2  =  \a2  {int.  Calc,  Art.  201),  and 
the  acceleration/  in  this  case  is  f  g  sin  i.  This  shows  that  the  velocity  of  the 
centre  of  gravity  of  the  cylinder  is  f  that  acquired  by  a  particle,  in  the  same 
time,  in  sliding  down  a  smooth  inclined  plane  of  the  same  inclination.  If  the 
cylinder  be  hollow,  k  =  a,  and  accordingly  f=\g  sin  i. 

2.  A  mass  M  draws  up  another,  M',  on  the  wheel  and  axle;  find  the  motion. 
Let  a  be  the  radius  of  the  wheel,  a'  that  of  the  axle  ;  then,  as  in  Ex.  3, 

Art.  133,  it  is  easily  seen  that  we  get 

l^pj  "(Ma°~  +  M'a"2-  +  fuW)  =  2g{Ma  -  M'a')  6  +  const. 

Hence,  by  differentiation, 

d29  _      g(Ma-M'a') 

d6 
Accordingly,  if  0  =  0,  and  —  =  0,  when  t  =  0,  we  get  for  the  angle  turned 
at 
through  in  the  time  t, 

x  Ma  —  M'a' 

e=*9t  Mtf+M'a't  +  frifi' 

3.  Find  the  tensions  of  the  strings  in  the  same  case. 

M'a  {a  +  a)  +  ^2       w    Ma  (a  +  a')  +  M2 
Am'  MgMa^M'a'*  +  nk*  9 Ma?  +  M' 'a'*  +  M2' 

4.  Find  the  velocity  acquired  by  the  centre  of  a  hoop  in  rolling  down  an_ in- 
clined plane  of  height  h.  -Ans.  sjljjlm 

135.  Work  done  by  an  Impulse. — If  a  mass  H 
moving  with  a  velocity  V  receives  an  impulse  in  the  direction 
of  its  motion,  and  if  Vf  be  its  velocity  after  the  impulse,  then 
the  change  in  its  kinetic  energy  is 

\M{J"-  F2)  =M(V'-  V).i{V'+  V). 

But  M{  V  -  V)  measures  the  impulse.  Hence  the  work 
done  by  the  impulse  is  measured  by  the  product  of  the 
momentum,  which  measures  the  impulse,  by  half  the  sum  of 
the  velocities  before  and  after  the  impulse. 

For  example,  a  bullet  m  in  passing  through  a  plank  expe- 
riences a  definite  amount  of  resistance,  measured  by  the 
thickness  and  by  the  resisting  force ;  but  this  equals  half  the 
loss  of  vis  viva  of  the  bullet,  or 

\m  (v~ -  vn)  =  m [v  -  v) .  \ [v  +  v), 


Compound  Pendulum. 


141 


where  v  and  v  are  the  velocities  with  which  it  meets  and 
leaves  the  plank.  Hence  the  momentum  m  (v  -  v)  commu- 
nicated to  the  plank  varies  inversely  as  v  +  v  :  consequently 
the  greater  the  velocity  of  impact  the  less  the  momentum 
imparted.  This  explains  how  a  bullet  with  a  high  velocity 
can  pass  through  a  door  without  moving  it  on  its  hinges. 

136.  Compound  Pendulum. — A  solid  body  oscillating 
under  the  action  of  gravity,  around 
a  fixed  horizontal  axis,  is  called  a 
compound  pendulum.  The  motion  of 
such  a  body  is  readily  reduced  to 
that  of  the  corresponding  simple 
pendulum,  as  follows  : 

Let  the  plane  of  the  paper  re- 
present that  in  which  the  motion  of 
G,  the  centre  of  inertia  of  the  body, 
takes  place,  and  let  0  be  the  point 
in  which  the  fixed  axis  intersects 
that  plane.  Draw  OY  vertically 
downwards,  and  let  GO  =  a,  M  =  mass  of  the  body, 
let  L  GOT  =  9. 

Suppose  the  pendulum  to  start  from  rest,  when  9  =  a ; 
then,  in  the  time  t,  the  point  G  will  have  descended  through 
the  vertical  height  a  (cos  9  -  cos  a) .  Also  the  vis  viva  of  the 
body  at  the  same  instant  (Art.  133)  is  represented  by 


Also 


;f)W, 


Hence,  by  the  principle  of  work,  Art.  132,  we  have 

/(— -)  =  2 Mga (cos  9  -  cos  a). 

If  the  moment  of  inertia  I  be  represented  by  MR2,  the 
latter  equation  becomes 

2ga  (cos  if  -  cos  o 


K- 


dt 


where  IT  is  the  radius  of  gyration  of  the  body  [Int.  Cede., 
Art.  197),  relative  to  the  axis  of  suspension. 


142  Work  and  Energy. 

Hence,  by  differentiation, 

$,£-•-«■  (13) 

Comparing  this  with  the  corresponding  equation  for  the 
motion  of  a  simple  pendulum  (Art.  101),  we  see  that  the 
motion  is  the  same  as  that  of  a  simple  pendulum  of  length 

a 

Again,  if  Mk2  be  the  moment  of  inertia  relative  to  an  axis 
through  the  centre  of  inertia  parallel  to  the  axis  of  suspension, 
we  have  {Int.  Calc,  Art.  196), 

K2  =  a2  +  k2 ; 

K2  k2 

hence  I  =  —  =  a  +  — .  (14) 

a  a  v     J 

The  point  0  is  called  the  centre  of  suspension.  If  OG  be 
produced  until  OC  =  I,  since  the  body  moves  as  if  its  entire 
mass  were  concentrated  at  the  point  C,  that  point  is  called 
the  centre  of  oscillation.  Again,  if  through  C  a  right  line  be 
drawn  parallel  to  the  axis  of  suspension,  all  the  points  of  this 
line  move  like  the  point  C,  i.e.  as  if  they  were  freely  sus- 
pended from  the  axis  of  rotation.  This  line  is  called  the  axis 
of  oscillation. 

Again,  since  OG  .  GO  =  k2,  the  axes  of  suspension  and 
oscillation  are  interchangeable,  i.  e.  the  time  T  of  an  oscilla- 

tion  is  the  same  for  both,  viz.,  T  =  tr  / . 

;  V    <*g 

By  varying  the  axis  of  suspension,  the  time  of  a  small 
oscillation  will  also,  in  general,  vary. 

For  parallel  axes,  T  obviously  is  a  minimum  when  a  =  k, 

and  the  corresponding  time  of  a  small  oscillation  =  tc    \ 

In  order  that  this  should  be  the  smallest  possible,  the 
axis  of  suspension  must  be  parallel  to  that  axis  round  which 
the  moment  of  inertia  is  least  {Int.  Calc,  Art.  217). 


Determination  of  the  Force  of  Gravity.  143 

If  the  axis  of  suspension  of  a  compound  pendulum  be 
inclined  at  an  angle  a  to  the  vertical,  it  is  readily  seen  that 
the  preceding  investigation  holds  good,  provided  g  sin  a  be 
substituted  for  g  throughout. 

Again,  as  in  Art.  101,  the  time  of  any  motion  of  a  com- 
pound pendulum  is  represented  by  an  elliptic  integral. 

Also,  if  a  solid  body  make  a  complete  revolution  round  a 
horizontal  axis,  the  time  of  revolving  through  any  angle  can 
be  reduced  to  that  for  the  corresponding  oscillatory  motion 
of  a  particle. 

Examples. 

1.  A  uniform  circular  plate,  of  radius  a,  makes  small  oscillations  about  a  hori- 
zontal tangent ;  find  the  length  of  the  equivalent  simple  pendulum.     Ans.  £  a. 

2.  Find  the  position  of  the  axis  with  respect  to  which  a  uniform  circular 
plate  will  oscillate  in  the  shortest  time. 

Ans.  The  axis  is  at  a  distance  of  half  the  radius  from  the  centre.    Length  of 
the  equivalent  pendulum  =  a. 

3.  Find  the  centre  of  oscillation  of  a  homogeneous  sphere,  of  radius  a,  oscil- 
lating round  a  horizontal  tangent  to  its  surface. 

Ans.  At  a  point  f  a  below  the  centre. 

4.  Find  the  ratio  of  the  times  of  oscillation  of  a  homogeneous  solid  sphere, 
and  of  a  spherical  shell  of  equal  diameter,  each  being  taken  with  reference  to  a 
horizontal  tangent.  j±ns%  -^21   :   5. 

5.  A  sphere  of  radius  a  is  suspended  by  a  fine  wire  from  a  fixed  point,  at  a 
distance  I  from  its  centre ;  prove  that  the  time  of  a  small  oscillation  is  repre- 

I5P  +  2a2 

sented  by  it   j — — (1  +  |sin2^a),  where  a  represents  the  amplitude  of  the 

vibration. 

6.  If  the  semiaxes  of  a  uniform  elliptic  disc  be  2  feet  and  1  foot,  and  it  be 
suspended  from  an  axis  perpendicular  to  its  plane  through  one  of  its  foci,  find 
the  time  of  a  complete  oscillation  under  gravity. 


Ans. 


V3 


\  9 


137.  Determination  of  the  Force  of  Gravity. — We 

have  already  seen  (Art.  103)  that  the  value  of  g  at  any  place 
can  be  determined  from  the  length  of  the  seconds  pendulum 
at  the  place.  To  apply  this  it  is  necessary  to  know  the  nu- 
merical value  of 


Two  methods  have  been  devised  for  this  purpose — one 
employed  by  Borda,  Arago,  Biot,  and  others ;  the  other  first 


144  Work  and  Energy. 

used  by  Bohnenberger,  and  afterwards  brougbt  to  great  per- 
fection by  Captain  Kater. 

In  the  first  method  the  compound  pendulum,  supposed 
made  of  a  material  of  uniform  density,  has  such  a  shape  that 
its  radius  of  gyration  can  be  calculated  mathematically,  as 
also  the  distance  of  its  centre  of  inertia  from  the  fixed  axis. 

The  second  method  depends  on  the  reciprocity  of  the 
centres  of  suspension  and  oscillation. 

Kater' s  compound  pendulum  consisted  of  a  heavy  bar 
having  two  apertures  at  opposite  sides  of  the  centre  of  inertia, 
through  which  knife  edges  passed,  on  either  of  which  the 
body  could  be  supported.  On  the  bar  was  placed  a  ring 
capable  of  being  moved  up  or  down  by  means  of  a  screw. 
Kater  moved  the  ring  until  the  times  of  oscillation  round  the 
two  axes  were  equal ;  in  which  case,  by  the  preceding,  the 
distance  between  the  axes  is  equal  to  the  length  of  the  equi- 
valent simple  pendulum.  The  distance,  /,  between  the  axes 
having  been  accurately  measured,  the  value  of  g  was  calcu- 
lated from  the  formula  g  =  — ,  where  T  denotes  the  time  of 

an  oscillation. 

Kater  published  an  account  of  his  observations  in  the 
Philosophical  Transactions,  1818,  1819.  For  a  more  detailed 
account  of  this  method  the  reader  is  referred  to  Bouth's  Rigid 
Dynamics,  Arts.  100-108. 

138.  Motion  of  a  Rigid  Body  round  a  Fixed  Axis. 

— In  general,  let  a  force  P,  in  a  direction  which  is  at  right 
angles  to  the  fixed  axis,  act  on  a  body;  then  for  a  small 
angular  motion  cl9  the  work  done  by  P  is,  by  Art.  128,  re- 
presented by  PpdO.  Again,  as  this  work  is  equal  to  the 
corresponding  change  in  the  kinetic  energy  of  the  body,  we 
have 

PpdQ  =  imzd(^j=  Ml/~  di 

Hence  we  get 

d~9  _  Pp      Moment  of  impressed  force  .-  -. 

d?=Mtf=         Moment  of  inertia  ^     ' 


W     ft       W+jF\ 


V 


Motion  of  a  Rigid  Body  round  a  Fixed  Axis.  145 

Examples. 

1.  A  uniform  circular  plate  of  1  foot  radius  and  1  cwt.  revolves  round  its 
axis  5  times  per  second  ;  calculate  its  kinetic  energy  in  foot  pounds.  \f 

Ans.  863,  approximately. 

2.  A  bent  lever  ACB  rests  in  equilibrium  when  AC  is  inclined  at  the  angle  o 
to  the  horizontal  line  ;  show*  that  when  this  arm  is  raised  to  the  horizontal  posi- 
tion it  will  fall  through  the  angle  2a,  C  being  supposed  fixed. 

3.  A  homogeneous  cylinder,  of  mass  M,  and  radius  a,  turns  round  a  hori- 
zontal axis ;  a  fine  thread  is  wrapped  round  it,  and  has  a  mass  M'  attached  to  its 
extremity.  Find  the  angular  velocity  of  the  cylinder  when  M'  has  descended 
through  the  height  h. 

A  2  _        ±M'0h 

KS'  W       a?{M+2M')' 

4.  A  right  cone  oscillates  round  a  horizontal  axis,  passing  through  its  vertex 
and  perpendicular  to  the  axis  of  the  cone ;  find  the  length  of  the  equivalent 
simple  pendulum. 

Ans.    — — — ,  where  h  is  the  height  of  the  cone,  and  b  the  radius  of  its  base. 
oh 

5.  If  in  the  last  example  the  cone  be  let  fall  from  the  position  in  which  its 
axis  is  horizontal,  find  its  angular  velocity  when  in  the  lowest  position. 

4A2  +  b2 

6.  In  the  same  case  find  the  pressure  on  the  fixed  axis,  at  the  lowest  position 
of  the  body,  arising  from  centrifugal  force  (Art.  98). 

Ans.   —  7T-75 — r^,  where  W  represents  the  weight  of  the  cone. 
2       4/r  +  0- 

7.  A  thin  beam,  whose  mass  is  M  and  length  2a,  moves  freely  about  one  ex- 
tremity attached  to  a  fixed  point  whose  distance  from  a  smooth  plane  is  b,  (b  <  2a) : 
the  other  extremity  rests  on  the  plane,  the  inclination  of  which  is  o.  If  the 
beam  be  slightly  displaced  from  its  position  of  equilibrium  determine  the  time 
of  its  small  oscillations. 

Indian  Civil  Service  Exam.,  1860. 

In  this  case  the  beam  may  be  regarded  as  turning  round  the  perpendicular 
on  the  plane. 

8.  A  bullet  weighing  50  grammes  is  fired  into  the  centre  of  a  target  with  a 
velocity  of  500  metres  a  second.  The  target  is  supposed  to  weigh  a  kilogramme, 
and  to  be  free  to  move.  Find,  in  kilogrammetres,  the  loss  of  energy  in  the 
impact. 

Lond.  Univ.,  1880.  Ans.  635'6. 

9.  "When  the  w-eight  P  of  the  pulley  is  taken  into  account,  show  that  equa- 
tion (9),  Art.  76,  becomes 


in  which  the  pulley  is  supposed  to  be  of  uniform  density  and  thickness. 

L 


146  Work  and  Energy. 

10.  If  the  motion  of  a  solid  body  acted  on  by  attracting  forces  be  a  pure  ro- 
tation, the  velocity  «  of  rotation  at  any  instant  will  be  given  by  the  equation 

JKF  («*-»>)  =  2  (F-  To), 

where  V  represents  the  potential  of  the  attracting  forces. 

11.  A  hollow  cylinder  rolls  down  a  perfectly  rough  inclined  plane  in  10  mi- 
nutes ;  find  the  time  a  uniform  solid  cylinder  would  take  to  roll  down  the  same 
plane.  Ans.  5  \/Z  minutes. 

12.*The  particles  composing  a  homogeneous  sphere  of  mass  M  and  radius  It 
were  originally  at  an  infinite  distance  from  each  other  :  find  the  work  done  by 
their  mutual  attraction. 

Suppose  the  sphere  in  question  to  have  been  formed  by  the  condensation  of 
an  indefinitely  diffused  nebula  ;  and  imagine  the  sphere  divided  into  a  number 
of  concentric  spheres.  Let  M'  be  the  mass  contained  in  the  sphere  whose  radius 
is  r  ;  then  we  have 

M'  =  M—  • 
IP 

Also,  if  dM'he  the  mass  bounded  by  the  spheres  r  and  r  +  dr,  then 

Accordingly  the  work  done  in  condensing  dM' ,  in  consequence  of  the  attraction 
of  the  interior  mass  M',  is,  by  (5)  Art.  126, 

fx  —  dM'  =3fi-^-  r*dr. 

Hence  the  whole  work  done  in  the  condensation  of  If  is 

Jf2 


M*  f*      7       3     M °~ 


(     147     ) 


CHAPTER    VII. 

CENTRAL    FORCES. 

Section  I. — Rectilinear  Motion. 

139.  Centre  of  Force. — We  next  proceed  to  consider 
motion  under  the  action  of  a  force  whose  direction  always 
passes  through  a  fixed  point,  and  whose  intensity  is  a  func- 
tion of  the  distance  from  that  point.  The  fixed  point  is 
called  the  Centre  of  Force;  and  the  force  is  said  to  be  attrac- 
tive or  repulsive  according  as  it  is  directed  towards  or  from 
the  centre. 

If  we  assume  that  two  particles  of  equal  mass,  placed  at 
the  same  distance  from  a  centre  of  attractive  force,  are  equally 
attracted  towards  the  centre,  when  they  are  conceived  placed 
together,  the  whole  force  acting  on  them — considered  as  one 
mass — will  be  double  that  which  acts  on  one  of  the  particles. 
Similarly,  if  any  number  (n)  of  equal  particles  be  placed 
together,  the  whole  force  will  be  n  times  that  which  acts  on 
a  single  particle.  Hence  it  follows  that  in  such  cases  the 
whole  attracting  force  is  proportional  to  the  number  of  par- 
ticles, i.  e.  to  the  mass  of  the  attracted  body — provided  the 
attracted  mass  be  of  such  small  dimensions  that  the  lines 
drawn  from  its  several  points  to  the  centre  of  force  may  be 
regarded  as  equal  and  parallel.  Accordingly  the  force,  in 
this  case,  is  proportional  to  the  attracted  mass ;  consequently 
the  acceleration  produced  by  it  is  independent  of  the  mass 
attracted,  and  is  a  function  of  the  distance  from  the  centre 
of  force  only. 

140.  Attraction. — The  acceleration  due  to  an  attractive 
force,  at  any  distance,  is  called  the  attraction  of  the  force, 
and  is,  as  we  have  seen,  independent  of  the  mass  of  the 
attracted  particle.  Consequently  the  measure  of  an  attractive 
force  at  any  distance  is  the  velocity  per  second  which  the 

L  2 


148  Rectilinear  Motion. 

central  force  could  generate  in  one  second,  in  its  own  direc- 
tion, if  it  were  conceived  to  act  uniformly  during  that  time. 
For  instance,  g,  i.  e.  the  velocity  acquired  in  one  second  by 
a  falling  body  (Art.  38),  measures  the  attractive  force  of  the 
Earth,  at  any  place,  and  is,  as  already  stated,  the  same  for 
all  bodies  at  that  place. 

141.  Rectilinear  Motion. — If  the  particle  acted  on  be 
originally  at  rest,  or  be  projected  in  the  line  joining  its  posi- 
tion to  the  centre  of  force,  its  motion  will  take  place  in  that 
right  line. 

Taking  this  line  for  the  axis  of  x,  and  the  fixed  centre  as 
origin,  we  have  for  the  equation  of  motion  (Art.  21), 

£--*•  (1) 

where  F  represents  the  attraction  at  the  distance  #,  which  is 
taken  with  the  negative  sign  because  it  tends  to  diminish  the 
velocity. 

We  shall  illustrate  equation  (1)  by  applying  it  to  a  few 
elementary  cases. 

142.  Force  Varying  as  the  Distance. — If  the  force 
be  proportional  to  the  distance  from  the  fixed  centre,  we 
may  assume  F  =  fix ;  then,  for  attractive  forces,  the  equation 
of  motion  becomes 

d*x  d2x  . 

^  =  -"*'or^+"*  =  0-  (2) 

This  equation  has  been  already  considered  in  Art.  109, 
and  accordingly  we  have 

x=  C  cos  t  yjl  +  C"  sin  t  y/fi.  (3) 

The  constants  C  and  C  are  determined  from  the  initial 
circumstances  of  the  motion. 

For  example,  if  the  particle  start  from  rest,  at  the  distance 
a  from  the  centre  of  force ;  then,  when  t  =  0,  we  have  x  =  a 

and  —  =  0  :    this  gies 

C  =  a,  and  C"=0; 


Inverse  Square  of  Distance,  149 

and  consequently  x  =  a  cos  t  ^/ju.  This  determines  the  posi- 
tion of  the  particle  at  any  instant,  and  shows  that  the  motion 
consists  of  a  simple  harmonic  vibration. 

Again,  if  (f  -  t)  ^//ul  =  2tt,  it  is  evident  that  the  values  of  x 

rfv 

and  of  —  are  the  same  at  the  end  of  the  time  f  as  at  the 

dt 
time  t :  this  shows  that  the  motion  is  oscillatory,  and  that  the 

time  of  a  complete  vibration  is  ——.     (Compare  Art.  111.) 

y/fl 

For  a  repulsive  force  the  equation  of  motion  is 

w-~  (4) 

Accordingly  (Art.  109),  we  have 

x  =  Ceu~*  +  Cfe~uK 

To  determine  the  constants  :  suppose,  as  in  the  former 
case,  the  particle  starts  from  rest,  at  the  distance  a ;  then 

a  -  C  +  C,  and  C  -  C  =  0. 

Hence  x  =  \a  [eu*  +  e*1*) .  (5) 

143.  Inverse  Square  of  Distance. — In  the   case  of 
the  law  of  nature,  in  which  the  attractive  force  varies  as  the 

inverse  square  of  the  distance,  we  have  F=  — ;  and  the  dif- 

x 

1  erential  equation  of  motion  is 

df      x2 

Multiplying  by  2dx,  and  integrating,  we  get 

'dxY    2M 

— 7 =  const. 

dt  J     x 


150  Rectilinear  Motion. 

Hence,  if  the  particle  be  supposed  to  start  from  rest,  at 
the  distance  a, 

This  equation  determines  the  velocity  at  any  distance 
from  the  centre  of  force. 

Again,  extracting  the  square  root,  and  transforming,  we 
get 

V^tf-.-^Lr-  (7) 


\x      a 


The  negative  sign  is  taken  since,  in  the  motion  towards 
the  centre  of  force,  x  diminishes  as  t  increases. 

To  integrate  this  equation,  assume  x  =  a  cos2  9 ;  then 


=  — — ,  and  dx  =  -  2a  sin  9  cos  9  cffl ; 

consequently  */%dt  =  2a%  cos2  9  d9  ; 

^-  (9  +  \  sin  29)  +  constant. 

Again,  the  constant  vanishes,  since  t  and  9  vanish  when 

x  =  a ; 

.:t  =  J^(0  +  isin20).  (8) 

Hence  the  time  of  motion  from  the  distance  a  to  the 
distance  x  is  •_ 

* = ii  (a  oos'1  J^" + v'*  («-*))•    (9) 

Also  the  time  of  motion  to  the  centre  of  force  is 
2  >J2/ 


Application  to  the  Earth.  151 

Again,  if  the  body  be  supposed  to  start  from  an  indefi- 
nitely great  distance  we  have,  making  a  =  go  in  (6), 

*>2  =  ^.  (10) 

iff 

144.  Application  to  the  Earth. — We  have  seen,  in 
Art.  126,  that  the  attraction  of  a  homogeneous  sphere  is  the 
same  as  if  its  mass  were  concentrated  at  its  centre.  Hence, 
the  results  of  the  last  Article  can  be  readily  applied  to  the 
approximate  determination  of  the  motion  of  a  body  falling 
from  any  height  above  the  Earth's  surface,  all  resistance  of 
the  atmosphere  being  neglected. 

In  this  case  g  measures  the  Earth's  attraction  at  its  sur- 
face ;  hence,  if  R  denote  the  Earth's  radius,  we  have  fx  =  gR2> 
and  if  this  value  be  substituted  for  /u,  we  can  readily  deter- 
mine the  velocity  and  time  of  motion  in  any  particular  case. 

For  instance,  the  velocity  Fwith  which  a  body  falling 
from  the  height  h  would  reach  the  surface  of  the  Earth  is 
given  by  the  equation 

Also,  by  (9),  the  time  of  motion  in  seconds  is 


IB  +  hLR  +  h   .    ,    I    h  \h  ) 


where  R  and  h  are  expressed  in  feet. 
If  R  =  nh,  this  becomes 


h  (1  +  n)  {  1  +  n    .     ,        1  , 

sm"1      ,  +  1 


When  n  is  a  large  number  this  becomes,  approximately, 

if  (•  *  k\ 


152  Rectilinear  Motion. 

If  the  body  be  supposed  to  start  from  an  infinite  distance, 
the  velocity  with  which  it  would  reach  the  Earth  is  given  by 
the  equation 

v*  =  2gR.  (13) 

145.  Comparison  of  Attraction  of  Different  Sphe- 
rical Bodies. — Let  If,  M'  denote  the  masses  of  two  spheres ; 
3,  £'  their  mean  densities  ;  r,  r  their  radii ;  f,f  their  attrac- 
tions at  their  surfaces,  respectively  :  then  we  have 

MM' 

For  example,  if  D  be  the  mean  density  of  the  Earth,  and 
R  its  radius,  then/,  the  attraction  at  the  surface  of  a  planet 
of  radius  r  and  mean  density  §,  is  given  by  the  equation 

f=9~-  (14) 

If  the  mean  densities  be  the  same  for  both,  we  have 

J  =  gR- 

If  we  assume  the  mean  density  of  the  Sun  to  be  one- 
fourth  that  of  the  Earth,  and  its  radius  104  times  that  of  the 
Earth,  then  the  velocity  acquired  in  one  second  by  a  falling 
body  at  the  Sun's  surface  is  approximately  represented  by  26g. 

In  the  case  of  the  mutual  attraction  of  two  spheres,  it  is 
often  convenient  to  assume  the  origin  at  their  common  centre 
of  gravity,  which  remains  a  fixed  point  during  the  motion. 
For  instance,  if  two  equal  spheres,  each  of  radius  r,  be  placed 
at  a  given  distance  apart,  and  left  to  their  mutual  attraction, 
this  method  may  be  employed  to  find  the  time  they  would 
take  to  come  together. 

Let  2a  be  the  initial  distance  between  their  centres,  and 
assume  the  origin  0  at  the  middle  point  of  the  line  joining 
the  centres.     If  x  be  the  distance  of  the  centre  of  either 

sphere  from  0  at  any  time ;  then  j-j  represents  the  corre- 


Examples.  153 

sponding  attraction,  and  the  time  required  is,  by  (9),  repre- 
sented by  the  expression 

where  fx  can  be  determined  by  the  equation 
r2     J     9  B  11  ' 


Examples. 

1.  If  k  be  the  height  due  to  the  Telocity  F"at  the  Earth's  surface,  supposing         / 
its  attraction  constant,  and  S  the  corresponding  height  when  the  variation  of 
gravity  is  taken  into  account,  prove  that 

1      J__J_ 
h~  H~  r' 

2.  If  a  man  weigh  10  stone  on  the  Earth's  surface,  calculate,  approximately,      \f 
his  weight  if  he  were  transferred  to  the  surface  of  the  Sun. 

Arts.  1  ton,  13  cwt. 

3.  Calculate,  approximately,  the  velocity  with  which  a  body  falling  from  an        ^ 
indefinitely  great  distance  would  reach  the  surface  of  the  Earth,  neglecting  all 
forces  besides  the  Earth's  attraction,  and  assuming  R  =  4000  miles. 

Ans.  7  miles  per  second. 

4.  Calculate,  in  like  manner,  the  velocity  with  which  a  body  falling  from  an 
indefinitely  great  distance  would  reach  the  surface  of  the  Sun. 

Ans.  364  miles  per  second. 

5.  In  a  work  erroneously  attributed  to  Sir  Isaac  Newton,  it  is  stated,  that  if 
two  spheres,  each  one  foot  in  diameter,  and  of  a  like  nature  to  the  Earth,  were 
distant  by  but  the  fourth  part  of  an  inch,  they  would  not,  even  in  spaces  void  of 
resistance,  come  together  by  the  force  of  their  mutual  attraction  in  less  than  a 
month's  time. 

Investigate  the  truth  of  this  statement.  Sch.  Ex.,  1883. 

Equation  (15)  gives  in  this  case  for  the  time,  in  seconds, 


sin-1  (-) 


49    .    ,  /1\  1 

—  sur1  [  - )  -f  — - 
90  \7/        8V3 


This  gives  about  5  minutes  and  38  seconds. 

If  the  question  be  solved  on  the  assumption  that  the  attraction  is  constant 
during  the  motion,  and  equal  to  that  when  the  spheres  are  touching,  the  time 
required  is  readily  found  to  be,  approximately,  =  100  VTl  =  5  m.  32  sees. 

It  may  be  observed  that  the  former  result  follows  from  this  immediately  by 
application  of  formula  (12). 


154  Rectilinear  Motion. 

6.  Show  that  if  a  sphere,  of  the  same  density  as  the  Earth,  attract  a  particle 
placed  at  the  nth  part  of  its  radius  from  its  surface,  the  time  of  motion  to  the 
surface  is  the  same  as  that  of  a  particle  moving  to  the  Earth  from  a  distance 
equal  to  the  nth  part  of  its  radius. 

7.  What  is  meant  hy  the  Astronomical  Unit  of  Mass  ? 

The  astronomical  unit  of  mass  is  that  mass  which  attracts  a_ particle  placed  at 
the  unit  of  distance  so  as  to  produce  in  it  the  unit  of  acceleration  in  the  unit  of 
time. 

8.  If  a  foot  and  a  second  be  taken  as  the  units  of  length  and  time,  calculate, 
approximately,  the  number  of  pounds  in  the  astronomical  unit  of  mass. 

Let  M  denote  the  mass  of  the  Earth,  and  m  that  of  the  astronomical  unit ;. 
then  we  have 

M  ,  M 

—  :  m  =  g  :  1,     or    m  =  —  , 
fi  »  gr* 

where  r  is  the  radius  of  the  Earth  in  feet.  Now  assuming  D,  the  mean  density 
of  tbe  Earth,  to  be  h\  tbat  of  water,  the  mass  of  a  mean  cubic  foot  of  the  Earth 
is  344  lbs.  approximately.  If  we  assume  the  radius  of  the  Earth  to  be  40  00  miles, 
we  get 

?L  =  _  —  x  344  =  951,000,000  lbs.,  approximately. 
gr>      3  g 

9.  Taking  the  value  of  gravity  as  981  in  centimetres  and  seconds,  and  the 
Earth's  radius  as  6-37  x  108  centimetres  :  find  the  Earth's  mass  in  astrono- 
mical units.  ^-ns.  398  x  1018. 

146.  Force  any  Function  of  Distance. — If  the  force 
be  attractive,  and  vary  inversely  as  the  nth  power  of  the  dis- 
tance, the  equation  of  motion  becomes 

Multiply,  as  before,  by  2dx,  and  integrate ;  then 


fdxY      2  in      1 
—  ]  =  — !-— — -  +  const., 
\dtj     n-lx71-1 


or 


2M     1 


n-1  xn~ 


+  const. 


If  the  attracted  particle  start  from  rest  at  the  distance  ctf 
we  have 

n-\  W'-1     a" 


>-*Tfi-  =  )-  (16) 


Elastic  Strings.  1,j5 

This  determines  the  velocity  at  any  distance  from  the 
centre. 

In  general,  if  F=  fi(jS(x)9  we  have 

and,  proceeding  as  before,  we  get 
'dx\2 


,  +  2ju  J  <p'(x)  dx  =  const. 
at 

If  F  denote  the  velocity  at  the  distance  a,  this  gives 

^-F2=2^(r,)-^(.r)j.  (17) 

If  the  body  start  from  rest  at  the  distance  a,  its  velocity 
at  any  distance  x  is  given  by  the  equation 

*-2,i  {*(«)-*(*)}■  (18) 

The  results  in  this  Article  follow  also  immediately  from 
Art.  131. 

147.  Elastic  Strings. — We  next  proceed  to  consider 
a  few  simple  cases  of  rectilinear  motion  for  heavy  bodies 
attached  to  elastic  strings. 

We  assume  that  in  all  cases  Hooke's  law,  that  the  tension 
of  the  string  is  proportional  to  its  extension  beyond  its  natural 
length,  is  applicable  throughout  the  motion ;  and  we  neglect 
the  weight  of  the  string. 

Let  us  commence  with  the  following  example  : — 

One  end  of  an  elastic  string  is  attached  to  a  fixed  point  on 
a  smooth  horizontal  table,  and  the  other  end  to  a  particle,  of 
mass  m,  on  the  table.  If  the  string  be  extended  beyond  its 
natural  length,  and  then  let  go,  to  find  the  subsequent  motion 
of  the  particle. 

Let  a  be  the  natural  length  of  the  string,  x  its  length  at 
any  instant  during  the  motion ;  then  x  -  a  represents  its 
extension  at  that  instant. 

Again,  let  b  represent  the  extension  when  we  suppose  the 
string  to  hang  freely  supporting  the  given  particle ;  then,  by 


156  Rectilinear  Motion. 

Hooke's  law,  the  tension  T  of  the  string  for  the  extension 
x  -  a  is  represented  by 

T=mgX-^.  (19) 

Accordingly,  the  equation  of  motion  of  the  particle  is 

Integrating,  we  have 

x  =  a  +  C  cos   /—  t  +  C  sin    —  t. 

To  determine  the  constants,  let  d  denote  the  initial  length 
of  the  string  ;  then 

a'  =  a  +  C,  i. e.  C  =  a' -  a; 

dx 
also,  since  —  =  0,  when  t  =  0,  we  have 


C"  =  0. 


(21) 


Consequently,  #  =  «  +  («'-  a)  cos   /—  ^. 

This  gives  the  position  of  the  particle  so  long  as  the  string 
is  stretched,  j.  e.  so  long  as  x  is  greater  than  a. 

The  velocity  at  any  instant  is  given  by  the  equation 


"(*-)Jf  ™Jy 


The  length  #  becomes  equal  to  #,  or  the  string  regains  its 
natural  length,  and  the  tension  ceases  to  act  at  the  end  of 


7T    fb_ 


the  time 

'  g 

Meanwhile   the  velocity  has  increased  from   zero,  and 
attained  its  maximum  value 


(«'-«)Jf 


at  the  same  instant. 


Weight  Suspended  by  an  Elastic  String.  157 

The  particle  will  now  continue  to  move  uniformly  along 
the  table  with  this  velocity  until  it  arrives  at  the  same  dis- 
tance a  on  the  opposite  side  of  the  fixed  extremity  of  the 
string,  when  it  becomes  again  acted  on  by  the  retarding 
tension  of  the  string  ;  and  the  same  motion  will  be  repeated. 

148.  Weight  Suspended  by  an  Elastic  String. — 
We  shall  next  consider  the  vertical  oscillations  of  a  body,  of 
weight  W,  attached  to  the  end  of  an  elastic  string,  which 
hangs  freely  from  a  fixed  point.  Suppose  the  body  de- 
pressed below  the  position  of  equilibrium,  and  then  set  at 
liberty,  to  investigate  the  subsequent  motion. 

As  before,  let  b  be  the  extension  of  the  string  due  to  the 
weight  W;  c  its  extension  at  the  commencement  of  the 
motion ;  x  its  extension  at  any  instant ;  T  the  corresponding 
tension  of  the  string  :  then,  by  Hooke's  law,  we  have 

T=W\,  (22) 

and  the  differential  equation  of  motion  is  obviously 

m^-W-T, 

|?  +  |  («  -  6)  =  0.  (23) 

The  integral  of  this  is 

x  =  b+C  cos    &t  +  C'sm^L 

To  determine  the  constants,  we  have,  when 

_  dx      A 
t  =  0,  x  =  c9  and  —  =  0  ; 

therefore,  C  =  c  -  b,  and  C  =  0. 

Consequently,     x  =  b  +  (c  -  b)  cos  Jj- 1.  (24) 

There  are  two  cases  to  be  considered,  according  as  c  is 
less  or  greater  than  2b. 

(1).  Let  c  <  2b.     In  this  case  the  extension  x,  and  con- 


158  Rectilinear  Motion. 

sequently  the  tension  T,  can  never  vanish  ;  and  the  body  will 
oscillate  up  and  down  through  the  distance  c  -  b,  on  both 
sides  of  the  position  of  equilibrium ;  the  time  of  an  oscilla- 
tion being  represented  by 


(2).  Next,  let  c  >  2b.     In  this  case  x  vanishes,  and  conse- 
quently T  also,  when 

b  +  (c-b)  cos  Jjt  =  0. 
The  corresponding  velocity  is  easily  found  to  be 


J 


go  {c  -  26) 


b 

As  the  tension  of  the  string  vanishes  at  this  instant,  the 
body  may  be  regarded  as  projected  upwards  with  the  fore- 
going velocity.  The  height,  h,  to  which  it  would  ascend  is 
given  by  the  equation 

*-£(«-»).  (25) 

The  body  will  afterwards  fall  to  the  origin,  and  the  subse- 
quent motion  will  be  as  before. 

149.  Weight  Dropped  from  a  Height. — Next  sup- 
pose the  weight  attached  to  the  string,  and  dropped  from  a 
height  h,  vertically  above  the  lower  extremity  of  the  string 
when  hanging  freely  and  unstretched.  The  solution  is  con- 
tained in  the  preceding  investigation  :  for  the  maximum  ex- 
tension c  of  the  string  is  given  by  (25),  and  is  represented  by 


c  =  b  +  yb{b  +  2h).  (26) 

In  practice  it  is  found  that  Hooke's  law  does  not  hold 
beyond  certain  limits  which  are  attained  long  before  the 
string  is  broken.  It  is  interesting  to  consider  whether  in 
any  particular  case  the  string  will  be  broken  or  not  by  the 
fall,  assuming  Hooke's  law  still  to  hold. 

A  given  string  is  capable  of  supporting  only  a  certain 
weight,  called  its  breaking  weight.     Denote  this  weight  by  B  ; 


Weight  Dropped  from  a  Height.  159 

then  e,  the  corresponding  extension  of  the  string,  is  found, 
by  Hooke's  law,  from 

W 

£=— e,  (27) 

and  the  string  will  break  or  not  according  as  the  maximum 
extension,  given  by  the  preceding  analysis,  is  greater  or  less 
than  e ;  that  is,  according  as  i  +  ^/b  [b  +  2h)  is  greater  or 
less  than  e. 

Again,  if  b  and  e  be  both  given,  the  least  height  of  fall,  h, 
in  order  that  the  string  should  break,  is  got  by  substituting  e 
for  c  in  (25),  and  is 

h  =  e±zm.  (28) 

Suppose  the  weight  W  to  be  the  nth  part  of  B,  i.  e.  let 
e  =  lib,  and  we  have  h  =  e  [%n  -  1). 

Thus,  for  instance,  a  weight  \  of  the  breaking  weight, 
dropped  from  the  height  e,  should  suffice  to  break  the  string. 

The  preceding  analysis  applies  also  to  the  vertical  oscilla- 
tions of  rods  supporting  heavy  weights ;  and  many  interesting 
practical  questions  are  explained  thereby — for  instance,  the 
danger  to  the  stability  of  a  suspension  bridge  arising  from 
the  steady  march  of  troops  over  it. — See  Poncelet,  Mecanique 
Industries,  'Arts.  332-345. 

Examples. 

1.  A  heavy  particle  attached  to  a  fixed  point  by  an  elastic  string  is  allowed 
to  fall  freely  from  this  point.  Show  that  the  elastic  force  at  the  lowest  point  is 
given  by  the  equation 

m         total  fall- 

F=  2W— : t-7-^j 

extension  of  string 

where  W  is  the  weight  of  the  particle. 

2.  A  heavy  particle  attached  to  a  fixed  point  by  an  elastic  string  hangs 
freely,  stretching  the  string  by  a  quantity  e.  It  is  drawn  down  by  an  addi- 
tional distance/;  determine  the  height  to  which  it  will  rise  if  p  -  e-  =  iae, 
a  being  the  unstretched  length  of  the  string.  Am.  2a. 

3.  A  heavy  body  is  attached  to  a  fixed  point  by  an  elastic  string,  which 
passes  through  a  fixed  ring,  the  natural  length  of  the  string  being  equal  to  the 
distance  between  the  ring  and  the  fixed  point. 

(a)  If  the  body  receive  an  impulse,  it  will  describe  an  ellipse  round  the  place 
it  would  occupy  if  suspended  freely. 

(h)  When  does  this  ellipse  become  a  right  line  ? 


160 


Central  Orbits. 


4.  A  particle  is  attached  by  a  straight  elastic  string  to  a  centre  of  repulsive 
force,  the  intensity  of  which  varies  as  the  distance  ;  the  string  is  at  first  at  its 
natural  length.  Find  the  greatest  distance  from  the  centre  of  force  to  which 
the  particle  will  proceed,  and  the  time  the  string  takes  to  return  to  its  natural 
length. 

5.  Two  bodies,  TFand  TV,  hang  at  rest,  being  attached  to  the  lower  end  of 
a  fine  elastic  string,  whose  upper  end  is  fixed :  supposing  one  of  them,  W ,  to  drop 
off,  find  the  subsequent  motion  of  the  other. 

Let  a  be  the  natural  length  of  the  string ;  b  its  extension  of  length  for  the 
weight  W;  c  that  for  the  weight  W ;  then,  at  the  end  of  any  time  t,  from  the 
commencement  of  the  motion  x,  the  depth  of  W  below  the  fixed  point  is  given 


by  the  equation    x  =  a  +  b  +  c  cos  t 


J 


6.  Two  particles,  connected  by  a  fine  elastic  string,  are  moving  in  the  direc- 
tion of  the  line  joining  them  with  equal  velocities,  their  distance  being  the 
natural  length  of  the  string ;  if  the  hinder  particle  be  suddenly  stopped,  find 
how  far  the  other  will  move  before  it  begins  to  return.  r  '■:  — 

Section  II. — Central  Orbits. 


'W 


150.  Plane  of  Orbit. — If  we  suppose  a  particle  acted 
on  by  a  force  directed  to  a  fixed  centre  to  be  projected  in  any 
direction,  it  is  easily  seen  that  its  subsequent  path  will  lie  in 
the  plane  passing  through  the  centre  of  force  and  the  direc- 
tion of  its  projection.  For,  since  the  force  acts  towards  the 
fixed  centre,  it  has  no  tendency  to  withdraw  the  particle  from 
that  plane  at  the  first  instant,  nor  at  any  subsequent  instant 
during  the  motion  ;  because  the  motion  of  the  particle  at  each 
instant  is  got  by  compounding  its  previous  motion  with  that 
due  to  the  central  force. 

We  shall  accordingly  take  this  plane,  called  the  plane  of 
the  orbit,  as  the  plane  of  rectangular  coordinate  axes ;  the 
fixed  centre  of  force  being  the  origin  0. 

151.  Differential  Equations  of  Motion. — Suppose 
the   force    attractive,  and   P  'the    y 

position  of  the  attracted  particle 
at  the  end  of  any  time  t. 
Let 

ON=x,  PN=y,  OP=r,  /_XOP=0. 

Suppose   F  to    represent    the 
acceleration  due  to  the  attractive  force 
have 


then,  by  Art.  68,  we 


Equation  of  Orbit,  and  Periodic  Time.  161 

(1) 


—  =  -i^cos  0  =  F- 
ar  r 


The  complete  determination  of  the  motion  for  any  law  of 
force  depends  on  the  solution  of  these  simultaneous  equations. 

In  the  case  of  a  repulsive  force  it  is  necessary  to  change 
the  sign  of  F. 

The  path  described  is  evidently  always  concave  to  the 
centre  of  force  for  attractive  forces,  and  convex  for  repulsive. 

152.  Law  of  Direct  Distance. — There  is  one  case 
in  which  the  differential  equations  can  be  immediately  inte- 
grated, viz.,  when  the  force  varies  directly  as  the  distanoe 
from  the  fixed  centre. 

Let  F=  fir ;  then,  for  attractive  forces,  we  have 

d2x  _  N 

(2) 


dt 


:jf  +  /«y  =  0 


The  integrals  of  these  equations,  by  Art.  109,  may  be 
written 

x  =  A  cos  t  y a  +  B  sin  t  a/ju  ) 

,-  /-  •  (3) 

y  =A'costyfi  +  B'smt^/n  ) 

The  arbitrary  constants  in  this,  as  in  all  other  cases,  can 
be  found  from  knowing  the  position,  velocity,  and  direction 
of  motion  at  the  first  instant. 

153.  Equation  of  Orbit,  and  Periodic  Time. — If 

we  solve  the  preceding  equations  for  cos  t  ^/p  and  sin  t  \Zfi> 
and  add  the  squares  of  the  results,  we  get 

(Ay  -  A'z)*  +  {By  -  B/x)2  =  (ABf  -  BA')7        (4) 

This  equation  represents  an  ellipse,  whose  oentre  is  at  the 
centre  of  force. 


162  Central  Orbits. 

Again,  if  2-nr  +  t*/ji  be  substituted  for  t^fx  in  equa- 
tions (3),  the  values  of  x  and  y  remain  unaltered ;  hence,  if 
(f  -  t)^/n  =  2tt,  the  body  will  occupy  the  same  position  at  the 
end  of  the  time  t'  which  it  occupied  at  the  time  t.  Accord- 
ingly, if  The  the  time  of  a  complete  revolution  in  the  orbit, 
we  have  o 

T  =  -— 

T  is  called  the  periodic  time,  and  is  the  same  for  all 
orbits  round  the  same  centre  of  force,  since  it  depends  only 
on  ju,  the  intensity  of  the  central  force,  i.e.  the  acceleration 
at  the  unit  of  distance,  and  not  on  the  initial  conditions  of 
the  motion. 

154.  Determination  of  the  Arbitrary  Constants. — 

Let  a,  b  be  the  coordinates  of  the  particle  at  the  instant 
from  which  the  time  is  reckoned,  V  the  initial  velocity,  and 
a  the  angle  which  the  initial  direction  of  motion  makes  with 
the  axis  of  x;  then,  making  t  =  0  in  equations  (3),  we  get 
A  =  a.  A  =  b. 

Again,  by  differentiation,  we  have 

—  =  B  y }x  cos  t  ^/ju  -  A  v/ju  sin  t  y/fx > 
ctt 

■j-  =  B^\/ijl  cos  t  y/fi  -  A'^/fx  sin  t »/ p. 

Hence  Fcos  a  =  B  ^//x,     Fsin  a  =  B'^/li  ; 

Fcos  a  ._  ~\ 


(5) 


consequently,    x  -  a  cos  t  v^u  +      ~7=~  sin  t  </ll 

Fsinq  [' 

y  =  b  cos  t  y  li  +       ~7=~  sin  t  \/  \x 

thus  the  position  of  the  particle  at  any  instant  is  determined. 
155.  Repulsive  Force. — Next,  if  the  force  be  repulsive 
the  equations  of  motion  are 

d?X  (Pi/ 


Several  Centres  of  Force.  163 

x  =  AetJ^  +  Be~u*  \ 
Hence,  as  before,  [  .  (6) 

y  =  Ae11*  +  B,e~t'^  ) 

If  we  solve  for  eu*  and  e~f \  and  multiply  the  resulting 
values,  we  get 

{Ax  -  Ay)  {By  -  Bx)  =  {AB  -  B'A)\  (7) 

This  represents  a  hyperbola,  having  the  lines 
Ax  -Ay  =  0,     By-  B'x  =  0 
for  its  asymptotes.     The  constants  A,  B,  A,  Bf  can  be  easily 
determined,  as  in  the  former  case,  whenever  the  initial  posi- 
tion, velocity,  and  direction  of  motion  are  given. 

Conversely  to  the  preceding  Articles,  it  can  be  readily 
shown  that  if  a  particle  describe  a  conic  under  the  action  of 
a  force  directed  to  its  centre,  the  force  varies  directly  as  the 
distance  ;  and  is  attractive  for  an  ellipse,  and  repulsive  for  a 
hyperbola. 

156.  Several  Centres  of  Force. — The  results  arrived 
at  above  hold  for  the  motion  of  a  body  acted  on  by  any 
number  of  centres  of  force,  each  varying  directly  as  the  dis- 
tance. For  it  is  readily  seen  that,  in  this  case,  the  forces  are 
equivalent  to  a  single  force,  directed  to  the  centre  of  mean 
position  of  the  different  centres  of  force,  whose  intensity  or 
absolute  force  is  equal  to  the  sum  of  the  intensities  of  the  diffe- 
rent centres  of  force  (see  Minchin's  Statics,  Art.  23). 

In  like  manner,  if  we  suppose  each  particle  of  a  body  to 
attract  according  to  the  law  of  direct  distance,  its  total  at- 
traction is  the  same  as  if  its  entire  mass  were  concentrated 
at  its  centre  of  inertia. 

Hence  it  follows  that  if  two  bodies  mutually  attract,  accord- 
ing to  this  law,  their  centres  of  inertia  will  describe  ellipses,  in 
the  same  periodic  time,  round  their  common  centre  of  inertia. 
This  result  holds  good  for  any  number  of  mutually  attract- 
ing bodies.  In  all  cases  the  path  described  by  the  centre  of 
inertia  of  a  body  is  called  the  orbit  of  the  body. 

Examples. 

1.  Prove  that  the  velocity  at  any  point  in  a  central  elliptic  orbit  varies  di- 
rectly as  the  diameter  drawn  parallel  to  the  tangent  at  the  point. 

M  2 


164  Central  Orbits. 

2.  In  the  case  of  a  repulsive  force,  varying  as  the  distance,  find  the  arbi- 
trary constants,  the  initial  conditions  being  supposed  the  same  as  in  Art.  154. 
Making         t  =  0  in  equations  (6),  we  get  a  =  A  +  B,  b  =  A'  +  B'. 
Again,  by  differentiation,  on  making  t  =  0,  we  get 


Hence, 


V  cos  a  =  {A  -  B)  V/*,      V  sin  a  =  {A' 


3.  Find  the  condition  that  the  orbit  in  the  preceding  should  be  an  equilateral 
hyperbola.  Ans.   V2  =  (a2  +  b2)  fi. 

4.  A  body  is  acted  on  by  four  equal  masses,  attracting  directly  as  the 
distance  ;  find  its  orbit,  and  show  that  its  periodic  time  is  one-half  of  that  of  a 
body  acted  on  by  one  of  the  masses  alone. 

5.  A  body  is  attracted  to  one  fixed  centre,  and  repelled  by  another,  of  equal 
intensity,  each  varying  directly  as  the  distance.     Find  its  path. 

Ans.  A  parabola. 

6.  In  the  ellipse  described  freely  by  a  body,  under  the  action  of  a  central 
force  varying  directly  as  the  distance,  determine  the  relation  connecting  the 
eccentric  angle  of  position  with  the  time  of  passage  through  any  point  on  the 
curve. 

7.  A  number  of  bodies,  which  describe  ellipses  about  the  centre  of  force  as 
centre  in  the  same  periodic  time,  are  projected  from  a  given  point  with  a  given 
velocity  in  different  directions  in  a  plane.  Prove  that  their  paths  will  all  touch 
a  fixed  ellipse  with  the  given  point  as  focus.   9338    Camb.  Math.  Trip.,  1875. 

8.  Being  given  the  centre  of  force,  a  point  in  the  orbit,  and  the  velocity  and 
direction  of  motion  at  that  point ;  give  a  geometrical  construction  for  the  lengths 
and  positions  of  the  axes  major  and  minor  of  the  orbit. 

We  now  return  to  the  general  equations  o:Tmotion  under 
Central  Forces. 

157.  Equable  Description  of  Areas. — In  equations  (1) 
if  the  first  be  multiplied  by  y,  and  the  second  by  x,  we  get  by 
subtraction 

d2y        d*x     _  d  f   dy        dx\      _ 

.___.  dy        dx 

Henoe  xJt~yTt  =  'h 

where  h  is  a  constant  independent  of  the  time. 


Velocity  at  any  Point.  165 

Again  (Art.  105,  Biff.  Calc),  we  have 
dy        dx      2d9 

xdi~ydi  =  r  dt; 

dd 
therefore  r2  —  =  h.  (8) 

at 

Hence,  if  A  denote  the  area  described  in  the  time  t  by  the 
radius  vector  r  drawn  to  the  particle,  we  have 


dA-±(r>dl 

dt  ~3V  dt 


7t)  =  ih; 


therefore  A  =  \{ht).  (9) 

No  constant  is  added  since  we  suppose  A  and  t  to  vanish 
together. 

If  we  suppose  t  =  1,  we  infer  that  h  is  double  the  area 
described  by  the  radius  vector  in  the  unit  of  time. 

Conversely,  if  a  particle  move  in  a  plane,  and  describe 
equal  areas  in  equal  times  around  a  fixed  point  in  the  plane, 
then  the  entire  force  acting  on  it  at  each  instant  passes  through 
the  fixed  point  (compare  Art.  28). 

158.  Velocity  at  any  Point,— Again  (Art.  183,  Biff. 
Calc),  we  have 

ds  _    od0 

pdt~rTt9 

where  ds  denotes  the  element  of  the  path  described  in  the 
time  dt,  and  p  is  the  length  of  the  perpendicular  from  the 
centre  of  force  on  the  tangent  at  the  point.     Hence 

ds  ds 

where  v  denotes  the  velocity  at  the  instant ;  therefore 

c=-.  (10) 

Accordingly  the  velocity  varies  inversely  as  the  perpen- 
dicular^. 


166 


Central  Orbits. 


The  constant  h  can  be  determined  from  (10)  whenever  the 
velocity  V,  the  distance  R,  and  the  direction  of  motion  at  any 
point  of  the  path  of  the  particle,  are  known. 

For,  let  $  denote  the  angle  which  the  direction  of  motion, 
at  the  instant,  makes  with  the  radius  vector  R ;  then  the  per- 
pendicular on  the  tangent  =  R  sin  0,  and  hence 

h=  VRsincp.  (11) 

Equation  (10)  admits  of  another  form;  for,  squaring,  it 
becomes 


v2 

P 

therefore 

v2- 

-'1**6 

where  u  = 

1 

r 

(W 

Cede, 

Art. 

183). 

(12) 

159.  Newton's  Proof. — On  account  of  the  importance 
of  the  preceding  results  we  shall  give  the  method  by  which 
the  equable  description  of  areas  was  originally  established  by 
Newton. 

Let  the  whole  time  be  divided  into  a  number  of  equal  in- 
tervals.   Then,  supposing 

no  force  to  act  on  the  body  w  y^  <z 

during  the  first  interval, 
it  would  describe  a  right 
line  AB,  uniformly,  -  in 
that  interval.  Likewise 
during  the  next  interval, 
if  no  force  act  on  it,  it 
would  describe  the  right 
line  Be,  in  the  direction  of, 
and  equal  to,  AB.  But  when  the  body  arrives  at  B,  suppose 
a  force  directed  to  8  to  act  on  it,  with  a  single  sudden  and 
great  impulse,  so  as  to  cause  the  body  to  deviate  from  the  right 
line  Be,  and  to  proceed  along  the  line  BC.  To  find  the 
position  of  the  body  at  the  end  of  the  second  interval,  we 
draw  from  c  the  line  cC  parallel  to  BS  (the  direction  of  the 
force),  and  meeting  BC  in  C;  then  the  body  will  be  found  at 
C  at  the  end  of  this  interval.     Join  SC  and  Sc ;  then,  since 


Velocity  at  any  Distance.  167 

SB  and  Cc  are  parallel,  the  triangle  SBC  is  equal  to  SBc,  and 
therefore  equal  to  the  triangle  SAB.  In  like  manner  D,E,  &c. 
the  positions  at  the  end  of  the  next  intervals,  can  be  deter- 
mined. Also  it  is  obvious  that  the  right  lines  AB,  BC,  CD,  &c, 
all  lie  in  the  same  plane,  and  the  triangles  SCD,  SDE,  &c., 
will  be  each  equal  to  SAB. 

Therefore  equal  areas  round  S  are  described  in  equal 
intervals  of  time ;  and,  componendo,  the  sum  of  the  areas 
described  are  proportional  to  the  time  of  their  description. 

If  now  we  suppose  the  number  of  intervals  of  time  in- 
creased, and  their  length  diminished  indefinitely,  the  path 
described  becomes  a  curved  line ;  the  centripetal  force  by 
which  the  body  is  perpetually  deflected  from  the  tangent 
to  the  curve  will  act  continuously ;  and  the  areas  described 
round  S9  being  always  proportional  to  the  time  of  their  de- 
scription, will  be  so  in  this  case  also. 

The  other  results  of  the  preceding  Article  follow  likewise 
(Newton,  Lib.  I.,  Sec.  n.,  Prop,  i.,  Principia). 

160.  Velocity  at  any  Distance. — In  equations  (1) 
if  we  multiply  the  first  by  2dx,  and  the  second  by  2dy, 
and  add,  we  get 

nd2x   ,       ^d2u  _  -^xdx+ydy         rt_7 

2  — -  dx  +  2-^  dy=-2F —  =  -  2Fdr. 

dt~  dt2  r 

Integrating,  we  get 

'dx\     (dy 


2  \Fdr  +  const. 
at  J       \dt  ' 

or  v2  =  -  2\Fdr  +  const.  (13) 

By  aid  of  this  equation,  when  the  law  of  force  is  given, 
the  velocity  at  any  point  in  the  orbit  can  be  determined. 

Thus,  let  the  acceleration  F  be  any  function  of  the  dis- 
tance represented  by  ju0'(r),  then 

v2  =  -  2/uj (j>'(r)  dr  +  const.  =  -  2/*0(r)  +  const. 

Again,  let  V  be  the  velocity  at  the  distance  B,  and  we 
get 

V2  =  -  2{i(j>lR)  +  const. ; 

therefore  r2  -  V2  =     2M  { 0  (R)  -  <f>  (r) ) .  (14) 


168  Central  Orbits. 

For  instance,  for  the  law  of  nature,  we  have 

*-r,-*G-i}  (15) 

Hence  we  see  that  the  velocity  at  any  distance  from  the 
centre  of  force  is  independent  of  the  path  described,  and  is 
the  same  as  if  the  body  had  been  projected,  with  the  initial 
velocity,  directly  towards  the  centre  of  force  (compare  Art. 
131). 

Again,  if  /=  — ,  we  have 

v*-  p.^SL/JL  _  JL\  (16) 

If  y  =  0,  when  R  =  <x>  ,  i.e.  if  the  velocity  at  any  point  in 
the  path  is  that  which  the  body  would  acquire  in  moving  from 
rest  from  an  infinitely  great  distance  towards  the  centre  of 
force,  we  have 

'--^4  (17) 

n-  1  rn  * 

For  instance,  if  the  force  vary  as  the  inverse  square  of  the 
distance,  we  have  in  this  case 

v>  =  &.  (18) 

T 

Again,  if  the  force  be  repulsive,  and  vary  directly  as  the 
nth  power  of  the  distance,  we  have  F  =  -  firn,  and  (14)  be- 
comes 

^  _  Vz  =  -^  (rn+1  -iZn+l).  (19) 

n  +  1 

If  F  =  0  when  i£  =  0,  i.e.  if  #?e  velocity  at  any  point  be  the 
same  as  that  acquired  in  moving  from  the  centre  of  force, 

2m    i*»  (20) 


+  1 


To  prove  the  Relation  F=-'^-.  169 

p*  dr 

161.  I*aw  of  Inverse  Square. — If  F=  ^,  equations  (1) 

v 

become 

s~e;  i-^}.  (2D 

1       f) 

Also  from  (8),  we  have,  —  =  -  ; 

/        /i 

hence,  equations  (21)  become 

h  r  h 


Integrating,  we  get, 


x  =  -  r  sin  9  +  a 


y  =     |  cos  0  +  P 


(22) 


in  which  a  and  /3  are  constants,  whose  values  can  be  found  by 
the  aid  of  the  initial  circumstances  of  the  motion. 

Again,  substituting  these  values  of  x  and  y  in  the  equa- 
tion xy  -  yx  =  h,  we  get 

tr  +(3x-ay-h  =  0.  (23) 

From  this  it  follows  that  the  orbit  is  a  conic  section  having 

the  centre  of  force  at  its  focus. 

Further  discussion  of  this  law  of  force  is  postponed  to 

Art.  166,  in  which  will  be  given  another  demonstration  that 

the  orbit  is  a  focal  conic. 

h2  dp 
162.  To  prove  the  Relation  .F=— -f . 

p?  dr 

Equation  (13)  gives,  by  differentiation, 

^.^=-14(1)^1  (24) 

dr  dr\p2J     p*  dr 


170  Central  Orbits. 

This  result  admits  of  a  useful  transformation ;  for,  if  7 
denote  the  semichord  of  curvature  drawn  through  the  centre 
of  force,  we  have 

y=pd£'       [Biff.  Calc, Art. 235.) 

Hence  the  previous  equation  becomes 

v2 
F=-.  (25) 

7 

This  result  can  also  be  readily  deduced  from  the  conside- 

ration  that  the  centrifugal  acceleration,  — ,  at  any  point  in  the 

orbit,  must  be  equal  and  opposite  to  the  component  of  the  cen- 
tral acceleration  taken  in  the  normal  direction  (Arts.  25,  90). 

Examples. 

1.  Prove  that  the  velocity  at  any  point  in  a  central  orbit  is  the  same  as  that 
acquired  in  moving  from  rest  along  one-fourth  the  chord  of  curvature  at  the 
point,  under  the  action  of  a  constant  force,  equal  in  intensity  to  that  of  the 
central  force  at  the  point. 

2.  A  particle  describes  a  circle  freely  under  the  action  of  a  force  whose 
direction  is  constant;  determine  the  law  of  force. 

Taking  the  centre  of  the  circle  as  origin  of  rectangular  axes,  the  axis  of  y 
being  parallel  to  the  constant  direction  of  the  force,  we  have 

dx  dx        dy 

dt         '        dt      y dt 


hence, 


dy  =_     x 
dt  ay 


^     d-y       a    I    du         dx\  a2  a2 

hence,  Y=  -£  =  —  (x-£  -  y —  \  = r* 

dfi       yz    \    dt         dt  J  yz 

3.  Apply  equation  (24)  to  find  the  law  of  force  directed  to  a  focus  in  an 
ellipse. 

In  this  case  we  have 


To  prove  the  Equation  -^r  +  u  =  ——0 .  171 

dO"  nrur 

1  dp       a    1        ,  _       ah3 

•"•     ir  =  7TTi     hence,  -F  =  -—  . 
ps  dr       b-  r-  b2r2 

4.  Find  the  law  of  force  in  the  curve 

rm  =  am  cos  Ww> 

Here  we  have  {Diff.  Calc,  Art.  190)  rw»Tl  =  amp. 

XT                                                 t,      («J+l)A3a2»» 
Hence,  F=  v ^ 

5.  Prove  that  the  force  under  whose  action  a  bodjT  P  revolves  in  any  orbit, 
about  a  centre  of  force  S,  is  to  the  force  under  whose  action  the  same  body  P 
can  revolve  in  the  same  orbit,  in  the  same  time,  round  another  centre  of  force  B, 
as  SP.  RP~  :  SG3,  where  SG  is  the  straight  line  drawn  from  S  parallel  to  PP, 
meeting  in  G  the  tangent  at  P  to  the  orbit.  Principia,  Sect,  n.,  Prop,  vii.,  Cor.  3. 

,7  2,.  Tp 

163.  To  prove  the  Equation  -^+w  =  T5— ?■ 
In  the  equation 

if  we  regard  r  as  a  function  of  0,  we  have 

-  2F-   ^    -  _  ill  ^M 
f/r  eft*    f/0 

Moreover,  from  (12),  we  have 

~W~M  dd[u  +  d¥J 

Substituting  in  the  preceding,  we  get 
d2u  F 

-77S  +  W  =  ; — ;  •  (26) 

dd2  h\u*  v     ; 

This  important  result  can  also  be  proved  as  follows  :  — 
Substituting  -  jFfor  P,  in  equation  (11),  Art.  28,  we  get 


Central  Orbits. 

*■-- 

d2r 
df  + 

(d9\\ 
'{dtp 

fdOV 

h%u\ 

by  (8) 

dr 
dt 

=  hu2 

dr  , 
ST"* 

d0 

172 

but 

also 

d2r  d  (du\  _        dO  d2u  _  a  <£u 

■'■   5»"""A5"A5eJ'"   dtaW" hudd*; 

consequently  F-  h2u2(-^-2  +  uj. 

The  discussion  of  central  orbits  comprises  two  distinct 
classes  of  questions.  In  the  one  it  is  required  to  find  the 
equation  of  the  orbit  when  the  law  of  force  is  known ;  in  the 
other  the  orbit  described  is  given,  and  the  law  of  force, 
directed  to  a  fixed  point,  is  required. 

In  the  latter  case,  if  the  origin  be  taken  at  the  fixed 

centre  of  force,  the  equation  of  the  orbit  can,  in  general,  be 

d2u 
expressed  in  terms  of  u  and  0,  from  which  the  value  of  -^ 

can  be  determined.    If  this  be  substituted  in  the  equation 

the  resulting  value  of  F  determines  the  required  law  of 
force. 

164.  Application  to  Ellipse. — For  example,  to  find 
the  law  of  force  which  will  cause  a  particle  to  describe  an 
ellipse  round  a  centre  of  force  situated  in  one  of  its  foci. 

Here  the  equation  of  the  orbit  is 

1  +  e  cos  9 

"  =  — L ' 

where  L  is  the  semi  latus-rectum. 


Laic  of  Inverse  Square.  173 


Hence 

d2n        e  cos  0 

d62~        L     ' 

therefore 

dhi      1 
U  +  dW  "  £' 

and  consequently 

7 

n_h2ui  _        h* 

L       a  (1  -  e2)  )• 


(27) 


Accordingly  the  force  varies  inversely  as  the  square  of  the 
distance  from  the  centre  of  force. 


Examples. 

Find  the  law  of  force,  directed  to  the  origin,  in  the  following  curves  :— 
1.     r  =  aead.  2.    u  =  aea9  +  be-a9.  3.     r  =  aea9  +  be'a9. 

Ans. 


Land  2.-.        3.*«£(~ ^-) 


165.  Case   where  the   I*aw  of  Force   is   given. — 

When  the  law  of  force  is  given,  the  determination  of  the 
orbit  depends  on  the  solution  of  a  differential  equation ;  for, 
if  F=fi<p(u),  equation  (26)  becomes 

^  +  U*$M.  (28) 

de*       h%  u2  v   ; 

This  equation  admits  of  being  completely  integrated  for  a  few 
laws  of  force  only.  We  shall  commence  with  the  most  im- 
portant case,  namely,  the  law  of  nature,  for  which  the  attrac- 
tion varies  as  the  inverse  square  of  the  distance. 

166.  Iiaw  of  Inverse  Square. — Let  F=-^  =  fxii29  then 

the  equation  becomes 

d2u  ix 

db%        v 


174  Central  Orbits. 

The  integral  of  this,  by  Art.  109,  is 

u  =  £  +  A  cos  (0  -  a).  (29) 

lb 

This  is  the  equation  of  a  focal  conic  (see  Art.  161). 

The  orbit  is  an  ellipse,  parabola,  or  hyperbola,  according 
to  the  values  of  the  constants  A  and  a.  These  constants  are, 
as  in  all  other  cases,  determined  from  the  initial  circumstances 
of  the  motion. 

We  commence  with  the  case  in  which  the  orbit  is  an 
ellipse. 

The  equation  of  an  ellipse  referred  to  a  focus  as  origin, 
and  to  any  line  drawn  through  it  as  prime  vector,  may  be 
written 

1       l+0COs(0-a) 
u  =  -  = 


r  a  (1  -  e%) 

Comparing  with  (29)  we  get 


H 


1  a 


h2      a  (1  -  e2)      b2i 
or  h2  =  n-  =  fiL.  (30) 

Hence,  in  different  orbits  round  the  same  centre  of  force,  h 
varies  as  the  square  root  of  the  latus  rectum. 

Again,  let  T  denote  the  periodic  time,  i.e.  the  time  in 
which  the  body  makes  a  complete  revolution  in  the  orbit ; 
then  since  h  represents  double  the  area  described  in  the 
unit  of  time,  we  have 

double  area  of  ellipse      2irab 

Hence,  from  (30), 

4ttV 


Kepler's  Laics.  175 

If  a  second  particle  be  supposed  to  describe  an  ellipse 
round  the  centre  of  force,  and  if  the  absolute  force  /j.  be  the 
same  in  both  cases,  we  have 

4ttV3 

where  a',  T'  are  the  semi-axis  and  the  periodic  time  in  its  orbit. 
Hence,  eliminating  fi,  we  get 


TV   fa\z 


rh 


(32) 


That  is,  the  squares  of  the  periodic  times  are  to  one  another  in 
the  same  ratio  as  the  cubes  of  the  semi-axes  major. 

167.  The  preceding  results  have  been  deduced  for  the 
motion  of  a  material  particle,  but  they  also  hold  good, 
approximately,  for  the  motion  of  the  centre  of  inertia  of  a 
body  of  finite  dimensions,  each  of  whose  elements  is  attracted 
towards  a  fixed  centre,  provided  the  dimensions  of  the  body 
are  small  in  comparison  with  its  distance  from  the  centre 
of  force.  For  in  this  case  the  attractions  on  the  several 
elementary  particles  of  the  body  may  be,  approximately,  re- 
garded as  a  system  of  equal  and  parallel  accelerations ;  and, 
consequently,  the  motion  of  the  body  will  (Art.  34)  be  the 
same  as  if  it  were  concentrated  at  its  centre  of  inertia.  Also, 
as  already  shown  in  Art.  126,  if  a  sphere  consist  of  homo- 
geneous spherical  strata,  its  entire  attraction  is  the  same  as  if 
its  entire  mass  were  concentrated  at  its  centre.  Accordingly, 
if  one  such  sphere  be  attracted  by  another  supposed  at  rest, 
its  centre  will  describe  an  ellipse,  having  the  centre  of  the 
attracted  sphere  for  a  focus. 

168.  Kepler's  Laws. — By  comparing  the  results  of  a 
large  series  of  observations  of  the  planets,  chiefly  of  Mars, 
made  by  Tycho  Brahe,  Kepler  arrived  at  the  following  laws 
concerning  the  planetary  orbits : — 

(1)  That  the  right  line  drawn  from  the  Sun  to  any  planet 
describes  equal  areas  in  equal  times. 


176  Central  Orbits. 

(2)  That  the  orbits  are  ellipses,  having  the  Sun  in  a  focus. 

(3)  That  the  squares  of  the  periodic  times  for  any  two 
planets  are  to  each  other  in  the  same  proportion  as  the  cubes 
of  their  mean  distances  from  the  Sun. 

From  the  first  of  these  laws  Newton  deduced  (Art.  157) 
that  each  of  the  planets  is  kept  in  its  orbit  by  the  action  of  a 
central  force  directed  to  the  Sun. 

From  the  second  he  proved  that  the  attractive  force  for 
each  planet,  in  its  different  positions,  varies  as  the  inverse 
square  of  the  distance  from  the  Sun  (Art.  164). 

From  the  third  law  he  deduced  that  the  absolute  force 
(fi)  is  the  same  for  all  the  planets  (Art.  166) ;  and  hence  that 
it  is  one  and  the  same  force,  directed  to  the  Sun,  by  which 
all  the  planets  are  retained  in  their  orbits.  These  laws  are 
only  approximate  when  we  take  account  of  the  mutual  actions 
of  the  planets  on  each  other  and  on  the  Sun. 

For  Newton's  demonstrations  the  student  is  referred  to 
the  Principia,  Lib.  I.,  Sect,  in.,  Prop.  xi. 

From  the  foregoing  we  infer  that  the  results  arrived  at 
for  the  motion  of  a  particle,  for  the  law  of  inverse  square  of 
the  distance,  are  applicable,  approximately,  to  the  planetary 
motions.  It  has  also  been  verified  by  observation  that  a  satel- 
lite belonging  to  any  planet  revolves  round  it  according  to 
the  same  laws  that  the  planets  revolve  round  the  Sun. 

169.  I^aw  of  Gravitation. — We  have  in  the  last  Ar- 
ticle given  a  brief  outline  of  the  process  by  which  Newton 
established  the  great  fundamental  law  of  attraction  of  matter, 
which  we  refer  to  as  the  law  of  nature,  and  which  may  be 
stated  as  follows  : — Every  particle  of  matter  in  the  solar  system, 
consisting  of  the  Sun,  the  planets,  comets,  Sfc,  exercises  on  every 
other  particle  an  attractive  force,  which  varies  directly  as  the 
product  of  the  masses  of  the  particles,  and  inversely  as  the  square 
of  their  mutual  distance. 

We  assume  that  this  is  a  general  property  of  matter,  and 
applies  to  all  matter  wherever  existing  in  the  universe.  This 
assumption  has  been  verified  by  observations  on  the  motion  of 
the  double  stars. 


Velocity  at  any  point  in  a  Focal  Orbit.  177 

170.  Expression  for  Velocity  at   any  Point  in   a 
Focal  Orbit. — We  commence  with  an  elliptic  orbit. 
In  this  case  we  have 

*-£-**    by(30), 

p-      ap-'      J  v     " 

where        p  =  SJY, 


the  centre  of  force  being  S. 

Again,  let  H  be  the  second 
focus  of  the  orbit,  SN,  HN'  per- 
pendiculars on  the  tangent  at  P,  the  position  of  the  particle. 

Suppose  HN'  =  p\   HP  =  r'\  then,    from  well-known 
elementary  properties  of  the  ellipse,  we  have 

r  +  r'=2a,    pp'=b\    £  =  -. 
p       r 

TTo«^  **     VPP'      HP       v-r'      fi(2a-ry 

xlence  tr  = =  —  =  — 

a  p*      a  p       a 

therefore  r  =  ^  -  £  (33) 

r       a  K     ' 

In  the  parabola  a  becomes  infinite,  and  we  have 

a  result  which  can  be  readily  established  independently. 

In  the  case  of  a  hyperbolic  path  we  have  r  =  2a  +  r,  and 
the  formula  becomes 

JJ-&  +  *  (35) 

r       a  v     ' 

Hence  we  infer  that  if  a  body  be  projected  with  a  velocity  Vy 

at  a  distance  R  from  the  centre  of  force,  the  orbit  described 

will  be    an   ellipse,    parabola,    or   hyperbola,    according   as 

T^o .  2/x 

V'  is  <  =  or  >  -£. 

This  result  may  be  exhibited  in  another  form  by  aid  of 
equation  (18),  as  follows  :  — 

The  velocity  at  any  point  in  an  ellipse  is  less,  in  a  para- 

N 


178 


Central  Orbits. 


bola  equal  to,  and  in  a  hyperbola  greater  than,  the  velocity 
which  the  body  would  acquire  in  moving  to  the  point  from 
an  infinitely  great  distance,  under  the  action  of  the  central 
force. 

171.  Construction  of  Orbit. — The  preceding  equation 
shows  how  to  construct  the  orbit 
when  we  are  given  the  absolute 
force,  the  initial  velocity,  position, 
and  direction  of  motion.  For, 
suppose  P  the  initial  position,  PT 
the  direction  of  motion,  and  S  the 
centre  of  force ;  let  V=  velocity  of 
projection,  SP  =  P  ;  then — 

(1)  if   V2  <  -jj  the  orbit  is  an  ellipse  whose  semi-axis  a 

is  given  by  the  equation 


a  R  fi 
Again,  draw  PIT,  making  the  angle  TPH  =  L  SPT, 
then  the  second  focus  H  lies  on  this  line,  and  its  position  H 
is  found  by  taking  PH  =  2a  -  P.  Consequently,  as  the  two 
foci  and  the  axis  major  are  known,  the  ellipse  is  completely 
determined. 


(2)  When  ^ 


V2  the  orbit  is  a  parabola,  which  can  be 


easily  determined  by  drawing  &ZV  perpendicular  to  the  direc- 
tion  of   motion   at  P,  inflecting 
ST=  SP,  and  dropping  NA  per- 
pendicular to  ST. 

The  parabola  described  with  S 
for  focus,  and  A  for  vertex,  will 
be  the  required  orbit. 

(3)  When  V2>\  the  orbit  is 

AX 

a  hyperbola,  whose  semi-axis  a  is  given  by  the  equation 


2_ 
B 


Effect  of  a  Sudden  Change  in  Absolute  Force.  179 

The  second  focus,  II,  can  be  easily  constructed,  as  in  the 
first  case,  but  lies  on  the  opposite  side  of  the  direction  of 
motion  from  the  centre  of  force  S. 

Again,  as  the  value  of  the  semi-axis  a  is  independent  of 
the  direction  of  projection,  we  infer  that  if  a  number  of 
bodies  be  projected  from  a  point  with  the  same  velocity,  in 
different  directions,  and  be  attracted  by  a  common  centre  of 
force,  the  mean  distances,  and  consequently  the  periodic 
times,  will  be  the  same  for  all  the  orbits. 

It  may  be  remarked  that  the  orbit  will  be  a  circle,  pro- 
vided the  angle  SPTis  right,  and  V2  =  ~  (compare  Art.  91). 

The  formulae  in  this  and  the  preceding  Article  are  of 
importance  in  the  discussion  of  focal  orbits.  We  add  a  few 
elementary  applications. 

Examples. 

1.  Calculate,  approximately,  the  periodic  time  of  a  planet  if  its  mean  dis-         v^ 
tance  from  the  Sun  is  double  that  of  the  Earth.  Ans.  1033  days. 

2.  If  a  body  be  projected  with  a  given  Telocity  about  a  centre  of  force 
which  varies  as  the  inverse  square  of  distance,  find  the  locus  of  the  centre  of 
the  orbit  described. 

Here,  since  the  locus  of  the  empty  focus  is  a  circle,  the  locus  of  the  centre 
is  also  a  circle. 

3.  In  the  same  case,  show  that  the  length  of  the  axis-minor  varies  directly 
as  the  perpendicular  drawn  from  the  centre  of  force  to  the  direction  of  pro- 
jection. 

Since  r  and  r'  are  each  constant,  p  is  to  p  in  a  constant  ratio  ;  consquently 
b  varies  as  p. 

4.  Show  that  there  are  two  directions  in  which  a  body  may  be  projected 
from  a  given  point  A,  with  a  given  velocity  V,  so  as  to  pass  through  another 
given  point  B. 

Since  the  axis-major  %a  is  given,  the  position  of  the  second  focus  is  deter- 
mined by  the  intersection  of  two  circles,  with  A  and  B  for  centres.  Hence  there 
are  two  solutions — one  for  each  point  of  intersection  of  the  circles. 

5.  Prove  that  the  time  of  describing  an  arc  of  a  parabolic  orbit,  bounded  by 
a  focal  chord  of  length  c,  varies  as  c*. 

172.  Effect  of  a  Sudden  Change  in  Absolute 
Force. — A  body  is  revolving  in  a  focal  orbit ;  if  when  it 
arrives  at  any  position  the  absolute  force  /j.  be  suddenly 
altered,  to  determine  the  subsequent  path. 

N  2 


180  Central  Orbits. 

Let  R  and  V  represent  the  distance  and  velocity  at  the 
instant  in  question,  and  let  ft  be  the  new  value  of  the 
absolute  force,  and  d  the  semi-axis  major  of  the  new  orbit ; 
then,  as  the  velocity  receives  no  sudden  or  instantaneous 
change,  we  have,  by  (33), 

2ft     n     2ft      ft 

R~a  =  -R~a"  (36) 

The  value  of  a\  and  consequently  the  position  of  the  new 
orbit,  can  be  immediately  determined  from  this  equation. 

For  example,  suppose  the  original  orbit  a  parabola,  and 
the  central  force  suddenly  doubled  in  intensity. 

Here  //  =  2/x,  and  our  equation  becomes 

2ft=4ft  _2p 
R  "  R       a'' 

hence  a'  =  R;  and,  consequently,  the  new  orbit  is  an  ellipse 
having  the  extremity  of  its  axis  major  at  the  point. 

If  the  change  in  ft  be  very  small,  and  represented  by  Aft, 
and  the  corresponding  change  in  a  by  £a>  it  is  plain  that  we 
have 

*—?*■  II- 3-  (37) 

Hence,  if  the  central  force  (or  the  attracting  mass)  be  in- 
creased slightly,  the  axis  major  will  be  diminished;  also,  if 
the  force  be  diminished  the  axis  major  is  increased. 

The  corresponding  change  in  the  periodic  time  is  readily 
found;  for,  by  (31),  we  have 

2  log  T  +  log  ft  =  2  log  2tt  +  3  log  a  ; 

2&T     3Aa      Aft 

hence  „     = ; 

1  a  ft 

therefore  —  =  -  -^  f—  -  1  J.  (38) 

Again,  if  the  centre  of  force  be  supposed  suddenly  trans- 
ferred to  a  new  position,  the  subsequent  path  can  be  readily 
constructed,  as  in  Art.  171. 


Application  of  Method  of  Hodograph,  181 

Examples. 

1-  A  number  of  bodies  are  projected  from  a  point  with  the  same  velocity,  but 
in  different  directions ;  prove  that  the  centres  of  their  orbits  are  situated  on  the 
surface  of  a  sphere. 

2.  A  body  is  describing  a  circle  under  a  central  force  in  its  centre  ;  if  the        V 
force  be  suddenly  reduced  to  one -half,  find  the  subsequent  path  of  the  body. 

Am.  a  parabola. 

3.  In  the  same  case,  if  the  central  force  be  suddenly  increased  in  the  ratio  of 
m :  1,  find  the  eccentricity  of  the  subsequent  path.  m  -  1 

Am.  . 

m 

4.  Two  equal  perfectly  elastic  particles  describe  the  same  ellipse  in  the  same 
period,  in  opposite  directions,  one  about  each  focus  ;  prove  that  the  major  axis 
of  the  orbit  is  a  harmonic  mean  between  those  of  the  orbits  they  will  describe 
after  impact. 

This  result  follows  immediately,  since  the  vis  viva  is  the  same  after  collision 
as  before  (see  Art.  81). 

5.  Prove  that  there  are  two  initial  directions  for  the  projection  of  a  particle 
with  a  given  velocity,  so  that  the  axis  major  of  its  orbit  may  coincide  in  direc- 
tion with  a  given  line. 


6.  If,  when  the  Earth  is  at  an  end  of  the  minor  axis  of  its  elliptic  orbit,  a 
meteor  were  to  fall  into  the  Sun,  whose  mass  is  the  mth  part  of  that  of  the  Sun  ; 
find  the  resulting  change  in  the  Earth's  mean  distance,  and  also  in  the  length  of 
the  year.  a  2T 

Am.  Aa  = ,      At  =  — — - . 

m  m 

173.  Application   of  Method  of  Hodograph.— The 

method  of  the  hodrograph  ("Art. 
26)  furnishes  a  simple  mode  of 
determining  the  law  of  force 
in  a  focal  ellipse.  For,  since 
the  velocity  at  any  point  P 
varies  inversely  as  the  perpen- 
dicular SL,  it  varies  directly  a' 
as  the  perpendicular  .fiTVdrawn 
from  the  second  focus ;  since  SL  x  HN  =  b°\ 

Consequently  the  hodograph  is  similar  to  the  locus  of  JV, 
when  turned  through  a  right  angle.  But  the  semicircle  de- 
scrihed  on  the  axis  major  as  diameter  passes  through  N,  con- 
sequently the  hodograph  is  a  circle. 

Again,  to  find  the  law  of  force,  let  Pi  denote  the  position 
of  the  movable  at  the  end  of  an  indefinitely  small  time  At, 


/ 


182  Central  Orbits. 

and  Ni  the   corresponding  position    of  N ;  then  (Art.  26) 
——  is  proportional  to  the  central  attractive  force. 

Join  the  centre  C  to  N  and  to  Ni ;  then,  by  an  elementary- 
property  of  the  ellipse,  CN  is  parallel  to  SP,  and  CN^  to 
fifPi. 

Let     SP  =  r,     lCSP=9,     SL=p,     fflST^p; 

then  lNCNx=lPSP,=  A0. 

.'  iVi^        A0     drA 

Also  (by 8),        _  =  „_  =  _. 

Hence  the  force  varies  inversely  as  the  square  of  the  dis- 
tance. 

Again,  since  v  =  -  =  —  p',  we  have 

"  V  ~KT  =    &a   *  r3 ' 

Consequently,  if  ,u  represent  the  absolute  force,  i.  e.  the 
force  at  unit  of  distance,  we  get 

_  tfa 

as  in  (30). 

Again,  since  the  velocity  at  P  is  proportional  and  per- 
pendicular to  jBTZV;  and  CN,  CH  are  constants,  it  follows 
that  the  velocity  at  P  can  be  resolved  into  two  constant  velo- 
cities— one  perpendicular  to  the  radius  vector,  the  other  to  the  axis 
major.         _  h 

Also,  since  the  velocity  at  P  is  represented  by  —  HN,  the 

component  velocity  perpendicular  to  SP  is  represented  by 
— ,  and  that  perpendicular  to  the  axis  major  by  —  :  i.e.  by 

j  and  j  e,  or  by    /—  and  J~  e,  respectively. 

That  the  hodograph  is  a  circle  in  this  case  appears  also  at 
once  from  (22).      For  if  x\   tj   be  the   coordinates  of  the 


Lambert's  Theorem.  183 

point  in  the  hodograph  which  corresponds  to  the  point  xy  in 
the  orbit,  we  have 

x'  ='x,  y  =  y ; 

hence,  substituting  in  (22).  and  eliminating  6,  we  get  for  the 
equation  of  the  hodograph 

(/-«)'+(//-/3y-  =  p 

which  is  the  equation  of  a  circle. 

We  may  here  observe  that  in  any  case  of  the  motion  of  a 
particle,  if  we  can  find  an  equation  connecting  the  velocities 
.r,  //,  z  of  the  motion,  with  constants,  that  equation  may  be 
regarded  as  that  of  the  hodograph,  in  which  x,  y,  z  are  the 
current  coordinates.     (See  Art.  26.) 

Example. 
A  particle  moving  in  an  ellipse  under  the  action  of  a  force  directed  to  a  focus 
has  a  small  velocity  n  y  impressed  on  it  in  the  direction  of  the  focus ;  find  the 
corresponding  changes  in  the  eccentricity,  and  in  the  position  of  the  apse. 

174.   Lambert's  Theorem.— In  Art.  140,  Int.  Calc,  it 
has  been  shown  that  the  area 
of  the  elliptic  sector  PSQ  is 
represented  by 


\ab  J0-0'-(sin0-sin^J),     ^ 

where  <p  and  q>  are  given  by  the  equations 

.     1         rfri+r2+c\$     .    1    ,     ./n  +  ra-cx" 

sm|f/)  =  |    ,    sin  | (p  =  j  ' 


in  which  SP  =  rlf  SQ  =  r2,  and  PQ  =  c. 

Accordingly,  if  t  represent  the  time  of  describing  the  arc 
PQ,  we  have 

.     2areaP/SQ      faz\h,         ,     ,.  .     ,a,        ,oon 

t  = =  (  - )  {^  -  <p  ~  (sm  <p  -  sin  0') } .      (39) 

This  shows  that  the  time  of  moving  from  any  point  P  to 
any  point  Q  can  be  expressed  in  terms  of  the  sides  of  the  tri- 
angle SPQ  and  of  the  axis  major  of  the  orbit. 


184  Central  Orbits. 

Again,  if  we  regard  a  as  becoming  infinitely  great  in  (39), 
we  get  for  t,  the  time  of  moving  from  P  to  Q  in  a  parabolic 
orbit, 

t  =  —  -  ( (n  +  r2  +  cf  -  (n  +  r2  -  *)*) .  (40) 

For  in  this  case  we  may  substitute  -  (  - )     for 

_  1  frx  +  r2  -  cVI 
0  -  sm  <£,  and  ^  I j    ior  0  -  sin  0  . 


Examples. 

1.  A  comet,  describing  a  parabolic  orbit,  being  supposed  to  cross  tbe  path  of 
the  Earth  ;  determine  tbe  points  of  ingress  and  egress  for  which  the  time  the 
comet  continues  witbin  tbe  Earth's  orbit  is  a  maximum. 

Ans.  The  extremities  of  the  axis  major. 

2.  Find  an  expression  for  the  time  in  the  preceding  question. 

2E 

Ans.   — ,  where  E  represents  the  length  of  the  year. 

3.  Two  planets,  describing  elliptic  orbits  in  a  common  period  round  the  Sun, 
being  supposed  to  pass  in  every  revolution  through  two  common  points  :  prove 
that  tbe  intervals  between  the  times  of  their  passage  through  the  points  are 
equal. 

175.  Modification  when  Mutual  Attraction  is 
taken  account  of. — The  preceding  investigations  are  based 
on  the  assumption  that  the  centre  of  force  is  fixed ;  accord- 
ingly they  can  be  applied  to  the  planetary  motions  only  on 
that  hypothesis.  However,  from  the  principle  of  the  equality 
of  action  and  reaction,  each  of  the  planets  exerts  on  the  Sun 
an  equal  and  opposite  attractive  force  to  that  which  the  Sun 
exerts  on  it.  We  proceed  to  consider  how  far  our  results 
must  be  modified  when  this  is  taken  into  account. 

We  have  seen,  in  Art.  13,  that  the  relative  motion  of  two 
bodies  is  unaltered  if  equal  and  parallel  velocities  be  given  to 
both.  We  accordingly  suppose  an  acceleration  applied  at 
each  instant  to  the  Sun,  equal  and  opposite  to  that  which  the 
planet  exerts  on  it ;  and  an  equal  and  parallel  acceleration 
applied  to  the  planet.  This  assumption  will  not  alter  their 
relative  positions,  while  it  reduces  the  position  of  the  Sun  to 


Modification  in  Kepler's  Third  Law.  185 

one  of  relative  rest.  Consequently  the  relative  motion  of  the 
planet  takes  place  in  the  same  manner  as  if  the  Sun  were  a 
fixed  centre  of  force,  and  the  planet  at  each  instant  were 
acted  on  by  the  sum  of  the  accelerations  that  the  Sun  exerts 
on  it,  and  that  it  exerts  on  the  Sun ;  since  these  accelerations 
take  place  in  opposite  directions  along  the  same  right  line. 

Again,  let  S  and  P  denote  the  masses  of  the  Sun  and 
planet  respectively  :  then  their  attractions  (being  proportional 

S  P 

to  their  masses)  may  be  represented  by  /—  and/—,  where  r 

represents  their  mutual  distance. 

Accordingly  the  total  acceleration  on  the  planet  towards 
the  Sun,  considered  as  a  fixed  centre,  is  represented  by 

AS  +  P) 

Consequently  in  our  preceding  investigations  we  must 
regard  the  absolute  force,  /z,  as  proportional  to  S  +  P  instead 
of  S ;  and  we  may,  by  proper  assumption  of  units,  take 

176.  Modification  in  Kepler's  Third  Law. — From 
what  has  been  just  established  it  follows  that  Kepler's  third 
law  is  only  approximate.  To  determine  a  more  exact  result 
we  must  substitute /(/S  +  P)  instead  of  /u,  in  (31),  for  one 
planet,  and/(#  +  P')  in  the  corresponding  formula  for  the 
other  planet,  when  we  have,  by  division, 

8  +  p-\ft)\Tj  (     > 

As  observation  shows  that  Kepler's  third  law  is  very 
nearly  exact  for  all  the  planets,  we  conclude  that  the  mass  of 
the  Sun  is  very  great  in  comparison  with  that  of  any  of  the 
planets.  In  fact  the  mass  of  Jupiter,  which  is  the  largest  of 
them,  is  less  than  a  thousandth  part  of  that  of  the  Sun. 

This  conclusion  will  appear  more  clearly  from  the  follow- 
ing method  of  comparing  the  mass  of  the  Sun  with  that  of  a 
planet  where  the  planet  has  a  satellite  : — 


186  Central  Orbits. 

177.   Comparison  of  Hasses  of  Sun  and  Planet. — 

Let  S  denote  the  mass  of  the  satellite,  $  its  distance  from  the 
planet,  t  its  periodic  time ;  then,  since  the  satellite  revolves 
round  the  planet  we  have,  as  in  last  Article, 


P+S      fSVfT 


S  +  P      \aj  \t 


\a 


(42) 


When  the  calculations  are  made,  it  is  found  that  in  all 
cases  [  - )  f  —  J  is  a  very  small  fraction  :  and  hence  also  — . 

If  S  be  supposed  very  small  in  comparison  with  P,  as  P  is 
in  comparison  with  S,  we  can,  by  (42),  obtain  the  ratio  of  the 
planet's  mass  to  that  of  the  Sun,  approximately. 

Again,  for  two  planets,  P  and  P',  if  the  masses  of  the 
satellites  be  neglected,  we  have 

p_  _  /sy  (C 
v  ~  \$)  U 

178.  Mass  of  Sun.— When  applied  to  the  Earth  and 
its  satellite  the  .Moon,  the  preceding  formula  gives  a  means 
of  comparing  the  mass  of  the  Sun  with  that  of  the  Earth. 

Let  E  and  31  represent  the  masses  of  the  Earth  and  the 
Moon,  r  their  distance,  then  equation  (42)  becomes 

S  +  E~\a)  \tj 

r        1 

Now,  as  a  rough   approximation,   we   assume  -  =  j^q  » 

i.  e.  that  the  Sun's  distance  from  us  is  400  times  that  of  the 

T 

Moon.     Also  we  take  —  =  13*4,  or  that  the  year  is,  approxi- 

0 

mately,  13*4  times  the  periodic  time  of  the  Moon. 

This  gives  |±|-  6Y709°5600  =  356>420  approximately. 

E 

Moreover,  as  determined  by  tidal  calculations,  M  =  ^  > 

hence  we  get  S  _  „fi-   .„~ 

E 


Mean  Demit ij  of  Sun.  187 

This  result  represents  very  closely  the  ratio  of  the  Sim's 
and  Earth's  mass  as  determined  by  more  exact  investiga- 
tions. 

The  foregoing  calculation  shows  the  enormous  mass  of  the 
Sun  in  comparison  with  that  of  the  Earth.  In  like  manner 
the  relative  masses  of  Jupiter,  Saturn,  and  other  planets 
which  have  satellites  can  be  found,  approximately. 


Examples. 

1.  Prove  that  the  mass  of  Jupiter  is  nearly  270  times  the  mass  of  the  Earth 

from  the  following  observations  : — Jupiter's  fourth  satellite  is  at  a  mean  distance         \^ 
of  25  radii  of  Jupiter,  and  its  periodic  time  is  16  days  18  hours;  Jupiter's  mean 
radius  is  11  times  the  mean  radius  of  the  Earth  ;  the  mean  distance  of  the  Moon 
is  60  radii  of  the  Earth,  and  a  mean  lunation  is  28  days. 

2.  Prove  that  the  mean  density  of  Jupiter  is  a  little  greater  than  that  of 
water,  and  that  the  mean  value  of  g  on  the  surface  of  Jupiter  is  about  71,  taking 
the  mean  density  of  the  Earth  as  5-67. 

179.  Mean  Bensity  of  Sun.— The  ratio  of  the  mean 
density  of  the  Sun  to  that  of  the  Earth  can  be  determined,  as 
follows : — 

From  (42)  we  have,  approximately, 


■m 


8 
E 

Again,  let  p,  px  denote  the  radii  of  the  Sun  and  Earth, 
and  o-  the  ratio  of  their  mean  densities;  then,  assuming  them 
spherical  bodies,  we  have 


K-gi 


8 
E  = 

'($ 

■<) 

m 

or 


-m- 


where  a  denotes  the  Sun's  mean  apparent  semi-diameter,  and  P 
the  Moon's  mean  horizontal  parallax. 


188  Central  Orbits. 

T 

If  we  substitute  16'  for  «,  and  57'  for  P,  and  take  —  as 

t 

before,  we  get  «r  =  0*23,  i.e.  the  Sun's  mean  density  is  about 

one-fourth  that  of  the  Earth. 

It  should  be  observed  that  this  result  does  not  require  a 

knowledge  of  the  Sun's  distance ;  and,  as  the  constants  in 

(43)  can  be  obtained  with  great  accuracy,  the  ratio  of  the 

mean  densities  of  the  Sun  and  Earth  can  be  determined  with 

great  precision. 

180.  Planetary  Perturbations. — The  previous  deduc- 
tions respecting  the  planetary  motions  are  only  approximate 
for  another  and  a  more  important  reason,  namely,  that  in  them 
we  have  neglected  the  mutual  actions  of  the  planets  on  each 
other. 

However,  since  the  Sun's  mass  is  very  great  in  comparison 
with  that  of  all  of  the  planets,  their  attractions  on  any 
member  of  the  solar  system  may  be  regarded  as  small 
disturbing  forces,  and  the  planetary  orbits  as  approximately 
ellipses. 

The  usual  method  of  treatment,  accordingly,  is  to  regard 
each  jolanet  as  moving  in  an  ellipse,  in  which  the  elements* 
are  subject  to  very  slow  changes,  arising  from  the  perturba- 
tions or  disturbing  effects  of  the  other  planets. 

In  this  manner  the  problem  has  been  discussed  by 
Lagrange,  Laplace,  and  other  great  writers  on  Physical 
Astronomy.  We  shall  not  enter  into  this  discussion,  as  it 
is  beyond  the  limits  contemplated  in  this  treatise.  There 
is,  however,  one  mode  of  considering  the  effects  of  a  disturb- 
ing force,  which  may  be  here  introduced.  This  consists  in  sup- 
posing the  disturbing  force  resolved  into  two  componentsf, 


*  The  elements  by  which  a  planet's  path  is  determined  are — (1)  its  mean  dis- 
tance from  the  Sun  ;  (2)  its  eccentricity;  (3)  the  longitude  of  its  perihelion;  (4)  the 
inclination  of  its  plane  to  a  fixed  plane;  (5)  the  angle  which  the  intersection  of 
these  planes  makes  with  a  fixed  line  ;  (6)  its  epoch,  or  the  instant  of  the  planet's 
being  in  perihelion. 

f  There  is  in  general  a  third  component,  perpendicular  to  the  plane  of  the 
orbit.  It  is  not  proposed  to  consider  the  effects  of  this  component  here.  This 
method  of  treating  the  disturbing  forces  is  discussed  in  a  masterly  and  lucid 
manner  by  Sir  John  Herschel,  in  his  Outlines  of  Astronomy,  ch.  12  and  13. 


Normal  Disturbing  Force. 


189 


one  along  the  tangent,  the  other  along  the  normal  to  the 
orbit,  and  in  treating  their  effects  separately. 

181.  Tangential  Disturbing  Force. — Suppose  P  the 
position  of  a  planet,  moving  in  the  ellipse BPA,  in  which  S  and 
H  are  the  foci ;  then,  since  a  tan- 
gential disturbing  force  alters  the 
velocity,  but  produces  no  effect  on 
the  direction  of  motion,  it  is  easy 
to  find  the  corresponding  changes 
in  the  elements  of  the  path.  For 
the  new  position,  H',  of  the  second 
focus  will  still  lie  on  the  line  PH. 

Again,  if  v  denote  the  velocity  at  P,  we  have,  as  before, 


2u 


When  the  change  in  v2,  caused  by  the  tangential  disturb- 
ing force,  is  known,  the  corresponding  change  in  a  can  be 
found ;  and  hence  the  position  of  H\  and  consequently  that 
of  the  new  axis  major. 

Thus  if  Ev  be  the  small  change  in  v,  due  to  the  disturbing 
force,  we  have 

2vhv  =  — r ; 


...     §a  =  —  v$v;         .-.   HH'=2Sa  =  —vSv.     (44) 
fx  /x 

If  the  tangential  force  act  in  the  direction  of  the  motion, 
and  consequently  increase  the  velocity,  a  will  also  be  in- 
creased, and  the  perihelion  A'  will  consequently  move 
towards  P. 

Again,  the  eccentricity  e  will  be  increased  when  811'  is 
greater  than  SH,  i.e.  when  P  is  between  the  perihelion  A 
and  the   extremity  of  the  latus-rectum  drawn  through  H. 

182.  Normal  Disturbing  Force. — Next,  if  a  normal 


190 


Central  Orbits. 


disturbing  force  act  at  P,  inwardly,  it  does  not  alter  the  velo- 
city, but  it  changes  the  direc- 
tion of  motion,  through  a  small 
angle  Sep.  As  the  velocity  is 
unchanged,  the  length  of  the  A 
semi- axis  major  a  is  unaltered,  a\ 
while  the  angle  SPTis  altered 
by  the  quantity  §<£.  Therefore 
the  angle  RPR',  between  PR  and  the  corresponding  line 
PR'  in  the  new  orbit,  is  280  ;  also  PR  =  PR'.  In  this 
manner  the  position  of  R'  is  found  when  the  angle  Btp  is 
known.  Again,  join  SR',  and  produce  it  at  both  ends,  then 
the  line  A'R'  will  represent  the  direction  of  the  axis  major 
of  the  new  orbit. 

Through  R  draw  DD'  perpendicular  to  SR.  The  points 
D  and  D'  are  called  the  quadratures  of  the  orbit.  When  P 
lies  between  D  and  the  perihelion  A,  the  line  AB,  called  the 
line  of  apsides  (see  next  Article),  moves  in  the  same  direction 
as  the  planet,  and  is  said  to  advance.  The  eccentricity  in- 
creases at  the  same  time.  If  the  planet  be  between  aphelion 
B  and  D,  the  eccentricity  continues  to  increase,  and  the  line 
of  apsides  recedes. 

Again,  in  moving  from  A  to  D' ,  the  disturbing  force  still 
acting  inwards,  it  is  easily  seen  that  the  line  of  apsides 
advances,  and  the  eccentricity  diminishes.  Hence,  in  the 
motion  from  quadrature  to  quadrature,  through  perihelion, 
the  apse  continually  advances,  in  the  case  of  a  normal  dis- 
turbing force  acting  inwards  ;  the  eccentricity  increases  during 
the  first  half  of  the  motion,  and  diminishes  during  the  second. 

The  contrary  effects  have  place  for  a  normal  disturbing 
force  acting  outwards. 

In  like  manner  in  the  motion  from  quadrature  to  quad- 
rature through  aphelion,  the  apse  recedes ;  the  eccentricity 
increases  during  the  first  half  and  diminishes  during  the 
second. 

183.  Apsides. — A  position  for  which  the  moving  body 
is  at  a  maximum  or  a  minimum  distance  from  the  centre  of 
force  is  called  an  apse.  The  corresponding  distance  from  the 
centre  of  force  is  called  an  apsidal  distance,  and  the  line  join- 
ing the  centre  of  force  to  an  apse  is  called  an  apsidal  line. 


Equation  for  Determination  of  A}) sides.  191 

Since  r,  and  consequently  u,  attains  a  maximum  or  a 
minimum  value  at  an  apse,  we  have  at  such  a  point 

^  =  0 

It  is  easily  seen  that  the  orbit  is  symmetrical  at  both  sides 
of  an  apse,  provided  the  force  is  a  function  of  the  distance 
only.  For,  if  a  particle  be  supposed  projected  from  a  point 
A  in  a  direction  perpendicular  to  the  line  OA  drawn  to  the 
centre  of  force,  it  is  obvious  that  for  the  same  velocity  of  pro- 
jection we  must  have  exactly  similar  paths,  whether  it  be 
projected  in  any  given  direction  or  in  that  exactly  opposite. 
Moreover,  if  the  velocity  were  reversed  at  any  point,  the  body 
would  proceed  to  describe  the  same  orbit,  but  in  an  opposite 
direction.  From  these  considerations  it  follows  that  the 
central  orbit  must  be  symmetrical  at  both  sides  of  an  apse, 
since  at  that  point  the  motion  is  perpendicular  to  the  central 
radius  vector. 

184.  An  Orbit  can  have  but  Two  Apsidal  Dis- 
tances.— For,  suppose  A  and  B  to  be  two  apsides,  and  the 
body  to  move  from  A  to  B ;  then  after  passing  B  it  will,  by 
the  preceding  Article,  describe  a  curve  similar  to  BA  ;  and 
so  on.  Hence  the  apsides  are  constantly  repeated,  and  the 
angle  between  two  consecutive  apsidal  distances  is  the  same 
for  all  positions  of  the  orbit.  This  angle  is  called  the  apsidal 
angle  of  the  orbit.  It  is  plain  that  a  central  orbit  cannot  be 
a  closed  curve  unless  the  apsidal  angle  is  commensurable  with 
a  right  angle. 

185.  Equation    for    Determination    of  Apsides. — 

Let  F=  fi(ji(u),  then  we  have,  by  (13), 


r  =  2fM 


W&  du  +  o, 


where  the  value  of  C  is  determined  by  the  initial  conditions  ; 

f(«) 


therefore         h*  ( u*  +  I  —  J  )  =  2fj. 


du  +  C.  (45) 


192  Central  Orbits. 

Hence,  as  -^  =  0  at  an  apse,  the  equation  for  determining 
the  apsidal  distances  is 

h'u^zJt^du+C.  (46) 

If  we  suppose  F=  ftu",  equation  (45)  becomes 

\\ddj  )       (n-l) 

and  the  equation  for  the  apsides 

tfu^-^rU^  +  C.  (48) 

n  -  1 

The  form  of  the  latter  equation  shows  that  it  cannot  have 
more  than  two  positive  roots,  which  therefore  correspond  to 
the  two  apsidal  distances. 

For  example,  let  the  force  consist  of  two  parts,  one  vary- 
ing as  the  inverse  square  of  the  distance,  the  other  as  the 
inverse  cube,  or 

F=nu*  +  fi'u>,  (49) 

then  h-  ir  =  2fiu  +  p'u*  +  C. 

Accordingly  the  apsidal  distances  are  in  this  case  deter- 
mined by  a  quadratic  equation.  If  ju  =  0,  there  is  but  one 
apsidal  distance. 

186.   Case  of  Velocity  due  to  an  Infinite  Distance. 

— The  integration  of  equation  (47)  in  a  finite  form  is  in 
general  impossible ;  there  is,  however,  one  case  in  which  the 
equation  of  the  orbit  can  be  readily  determined,  viz.,  when 
the  velocity  at  any  point  is  that  acquired  in  moving  from  an 
infinite  distance  under  the  action  of  the  central  force. 


For  we  have,  m  this  case,  by  (17),  r  = -  u 


B-I 


therefore  *  +  (gj.-3fa«».  (50) 


Case  of  Velocity  due  to  an  Infinite  Distance.  193 

Hence        ~jz  =  u  y/aun~*  -  1,  writing  a  instead  of 


dO  '  (n-l)h2' 

therefore  0  = 


J  u  yaun~3  -  1 
To  integrate  this,  let  aun~3  =  — ,  then  —  -  - 


du 

and  we 


get     f- 
J  u 


u  n  -  3 

dz 


v/W1"3-!         n-Z)</i-z 


2 

-  cos_12  +  const. ; 


r>  q 

.\     0  +  (5  =  — — -o  cos"^,  or  s  =  cos  — —  (0  +  j3), 
where  /3  is  an  arbitrary  constant : 


hence  r~  =  ij ^  cos  ^-? (0  +  /3).  (51) 

If  «  denote  the  apsidal  distance,  and  0  be  measured  from 
the  apsidal  line,  the  preceding  may  be  written 

«-3       n-3        n  _  3 
r  2  =  «  s   cos  _^  0.  (52) 

This  is  the  polar  equation  of  the  orbit. 

For  example,  when  n  =  2,  we  get  the  parabola 

rz  cos  i  9  =  ah. 

Again,  when  ^  =  5,  it  becomes 

r  =  a  cos  0 ; 

a  circle  having  its  centre  on  the  circumference. 

o 


194  Central  Orbits. 

For  n  =  7  we  get  the  lemniscate 

r  =  «2  cos  20, 
and  so  on. 

Equation  (52)  fails   when  n  =  3 ;  in   this   case,  however, 
(50)  becomes 

which  gives  £0  =  log  w  +  const.,  where  A-  =    /—  -  1, 


or 


u  =  Beko. 


This  is  the  equation  of  a  logarithmic  spiral. 

187.  Approximately  Circular  Orbits.— If  the  orbit 
described  round  a  centre  of  force  be  nearly  a  circle,  its  equa- 
tion can  be  found  approximately,  as  follows : — 

Assume  F=  fiu2f(u),  then  equation  (26)  becomes 


+ 


« =  &/(«). 


d6*  K 

If  the  orbit  were  an  exact  circle  we  should  have 

A  dh>       ft 


therefore  a  must  satisfy  the  equation 


a  =  £/(«).  (53) 


When  the  orbit  is  approximately  circular  we  may  assume 
w  =  #,  +  z,  where  s  is  always  very  small. 

Hence  ^  +  a  +  s  =  ^/(«  +  s), 

or  Cdi2+a  +  Z==tf  ^  +  ^^ ' ' 


Approximately  Circular  Orbits.  195 

By  (53)  this  becomes,  neglecting  s2  and  higher  powers  of  s, 

8-(l-£/'M)  =  0; 

or,  substituting  — -  for  £, 

If     Jx  =  1  -  — — ,  this  becomes 
./  («) 

g  +  fa  =  0.  (54) 

When  k  is  positive,  the  integral  of  this,  by  Art.  109,  is 

Z  =  C  COS  [By/li  +  a), 

or  u  =  a  +  c  cos  (0  y^  +  a),  (55) 

when  c  and  a  are  arbitrary  constants. 

The  greatest  value  of  u  is  a  +  c ;  consequently,  in  order 
that  the  orbit  should  be  approximately  circular,  it  is  necessary 
that  c  should  be  very  small  in  comparison  with  a. 

Again,  supposing  c  positive,  the  greatest  value  of  u  has 
place  when  9  */k  +  a  =  0,  and  the  least  when  6  x/k  +  a  =  it  ; 
consequently  the  apsidal  angle  is 

7T  IT 

or 


If  k  be  negative,  i.e.  if  — —  >  1,  the  integral  of  (54)  is 

of  the  form 

z  =  Aee'~k  +  Be-6^, 

and  therefore  z  would  either  increase  or  diminish  indefinitely 

02 


196  Central  Orbits. 

with  0 ;  and  accordingly  the  orbit  cannot  be  approximately 
circular  in  that  case. 

The  value  of  k  depends  on  the  law  of  force  :  for  example,  if 
the  force  vary  inversely  as  the  nth  power  of  the  distance,  then 

/(«)  =  „«•-*,     and    ^W-»-2. 

Accordingly,  in  this  case,  k  =  3  -  n. 

Hence  a  nearly  circular  orbit,  having  the  centre  of  force  in 
the  centre,  is  impossible  for  laws  of  force  which  vary  inversely  as 
a  higher  power  than  the  cube  of  the  distance. 

When  n  is  less  than  3,  the  angle  between  the  apsides  is 


y3-n 

For  instance,  if  n  =  2,  the  angle  is  tt  ;  this  agrees  with 
what  has  been  already  proved,  as  the  orbit  is  a  focal  conic  in 
this  case. 

Again,  if  n  =  -  1,  the  angle  is  \ir,  as  it  ought  to  be,  since 
the  orbit  is  a  central  ellipse. 

188.  Movable  Orbits. — If  a  central  orbit  be  made  to 
move  in  its  own  plane  with  an  angular  velocity  propor- 
tional at  each  instant  to  that  of  the  radius  vector  in  the  orbit, 
we  can  easily  show — (1)  that  the  new  orbit  is  also  a  central 
orbit ;  (2)  that  the  difference  between  the  forces  in  the  two 
orbits  varies  inversely  as  the  cube  of  the  distance  from  the 
centre  of  force.  (Newton,  Principia,  lib.  i.,  sect.  9.) 

In  a  central  orbit  we  have,  in  general, 

<M  <Pr  _   t  (d&\2_  p 

oY  ~  r  \dt  J  ~     ' 

where  k  is  constant,  the  former 


r alt 

=  h, 

an( 

If 

now  we 

make  9  = 

w 

equatii 

Dn  gives 

r 

dt 

h 
~k 

-  =  -  =  h  (suppose). 

This  shows  that  the  point  describes  equal  areas  in  equal 
times  round  the  origin ;  accordingly  the  new  path  described 
by  the  point  is  also  a  central  orbit. 


Examples.  197 

Again,  the  second  equation  may  be  written 

d'-r        (dPs*  /W 

dJ-r\-dt)  =  P-[1;-l]'\dt 

h2  -  h'2 

=  P  + —  =  P'  (suppose) ; 

h2  -  Ji2 
hence  P'  -  P  = —  :  this  shows  that  the  difference  be- 

.  r  .  1 

tween  the  forces  in  the  fixed  and  movable  orbits  varies  as  — . 

Hence,  from  any  central  orbit  we  can  get  another,  called 
by  Newton  a  revolving  orbit ;  and  the  equation  of  the  revolv- 
ing orbit,  in  polar  coordinates,  is  derived  from  that  of  the 
original  by  substituting  kQ  for  6 ;  where  the  constant  k  is 
determined  from  the  initial  conditions. 

For  example,  when  F  =  — ,  we  get  a  focal  conic,  whose 

equation  is  of  the  form 

A 

f  = • 

1  +  e  cos  (9  -  a)  ' 
hence,  if  F=  —  +  — ,  the  equation  of  the  orbit  is  of  the  form 


1  +  ecos{kd  -  a) 


The  apsidal  angle  in  the  new  orbit  is  equal  to  that  in  the 
original  orbit  divided  by  k,  as  is  readily  seen.  Newton  applied 
this  method  to  the  investigation  of  the  apsidal  angle  in  the 
lunar  orbit.  His  discussion  is  beyond  the  limits  proposed  in 
the  present  treatise.  Moreover,  the  progression  of  the  Moon's 
apse,  thus  determined  by  Newton,  is  but  half  its  true  amount. 

Examples. 

1 .  Find  the  law  of  force  in  a  circle  when  the  centre  of  force  is  situated  on 
its  circumference.  1 

Ans.     — . 

2.  Investigate  the  motion  of  a  hody  which  is  acted  on  hy  several  centres  of 
force  varying  directly  as  the  distance ;  and  show  how  to  construct  the  position 
of  the  centre  of  the  orhit. 


198  Central  Orbits. 

3.  In  the  same  case,  find  the  condition  that  the  orhit  should  he  a  parahola. 

4.  Assuming  ihat  the  law  of  force  in  a  focal  conic  is  that  of  the  inverse 
square  of  the  distance,  show  that  the  converse  theorem  can  he  immediately 
established,  viz.,  that  a  particle  attracted  by  a  centre  of  force,  varying  accord- 
ing to  that  law,  will  describe  a  conic,  having  the  centre  of  force  in  one  of  its 
foci. 

5.  A  semi-ellipse  is  freely  described  by  a  particle  under  the  action  of  a 
force  parallel  to  its  axis  of  figure  ;  determine  the  requisite  law  of  force,  with 
the  velocity  of  the  particle  on  reaching  or  leaving  either  extremity  of  the  semi- 
ellipse. 

6.  Prove  that  the  law  of  force  in  an  equiangular  spiral  is  that  of  the  inverse 
cube  of  the  distance  ;  and  explain  why  we  cannot  assert,  conversely,  that  a  body 
acted  on  by  such  a  force  will  describe  an  equiangular  spiral. 

7.  If  the  velocity  at  each  point  in  a  central  orbit  be  equal  to  that  in  the 
equidistant  circle,  prove  that  the  orbit  is  an  equiangular  spiral  for  an  attractive 
force.  

By  Art.  91  the  velocity  in  the  equidistant  circle  =  "s/Fr.    Again,  by  Art.  162, 

the  velocity  in  the  orbit  =  -Jpy  •    therefore  r  =  y  =  p  — .     Hence  —  =  —  ; 
J  ' '  'dp  r       p 

therefore  r  =  kp,  and  consequently  the  orbit  is  an  equiangular  spiral. 

If  the  force  be  repulsive,  the  orbit  is  an  equilateral  hyperbola. 

8.  In  general,  if  the  velocity  at  each  point  in  a  central  orbit  be  in  a  constant 
ratio  to  that  in  an  equidistant  circle  ;  find  the  law  of  force  and  the  equation  of 
the  orbit.  _ 

Let  the  constant  ratio  be  represented  by  1  :  Vn;  then,  as  in  preceding 
example,  we  have 

r  =  np  —  ;  hence  p  =  krn. 
dp 

From  this  it  is  easily  seen,  as  in  Art.  162,  that  the  equation  of  the  orhit  is 
of  the  form 

r"-1  =  a""1  cos  (n  -  1)  0. 

The  law  of  force  is  readily  found  ;  for,  in  general, 
1  1  1 

F  QC   — —  OC    —r~   OC    — • 

p2y       p-r        r~"+1 

9.  In  the  same  case,  show  that  the  velocity  at  each  point  in  the  orbit  is  that 
due  to  motion  from  an  infinite  distance,  subject  to  the  central  force. 

Here  v°-  =  — ■  =  -£—  ;  hence,  by  (17)  Art.  160,  the  velocity  is  that  due  to 
n        nr'in 


an 


infinite  distance. 


10.  When  the  velocity  and  direction  of  motion  at  any  point,  as  well  as  the 
centre  and  intensity  of  the  force,  are  given,  show  how  to  find  the  radius  of  cur- 
vature of  the  orbit  at  the  point. 

11.  A  body  is  acted  on  by  two  attractive  centres  of  force,  of  equal  intensity  ; 
and  also  by  a  repulsive  force  from  another  centre,  of  double  the  intensity;  the 


Examples.  199 

forces  varying  directly  as  the  distance.  Prove  that  the  orbit  is  a  parabola,  and 
show  how  to  construct  its  focus  and  directrix  when  the  initial  velocity  and  di- 
rection of  motion  are  given. 

12.  If  a  repulsive  force  vary  as  the  inverse  square  of  the  distance,  prove 
that  the  orbit  is  a  branch  of  a  hyperbola,  having  the  centre  of  force  in  the  focus 
external  to  the  orbit. 

13.  A  particle  is  acted  on  by  a  central  repulsive  force,  which  varies  as  the 
nth  power  of  the  distance.  If  the  velocity  at  any  point  be  that  due  to  motion 
from  the  centre  of  force ;  find  the  equation  of  the  path. 

Here,  by  (20),  Art.  160,  we  have 

.         2^      _,     ...    1  2f,         _ 


»  +  1        '        f      (n  +  1)  A2 
\dd)       w*+1  h2{n+  1) 


therefore 


hence  we  get 


tt+l 

die 

-  dfl 

V/fc2- 

-  e<«+3 

n+3 

u  2  = 

n 
a  cos  - 

+  3 

14.  If  the  velocity  at  each  point  in  a  central  orbit  varies  directly  as  the  dis- 
tance from  the  centre  of  force,  prove  that  the  orbit  is  an  equilateral  hyperbola, 
and  find  the  law  of  force. 

15.  Show  that  the  velocity  at  any  point  in  a  focal  parabola  is  to  that  in  the 
equidistant  circle  as  V2  :  1 . 

16.  If  the  law  of  force  be  that  of  the  inverse  cube  of  distance,  investigate 
the  different  varieties  of  orbit  described. 

Let  F  =  /jm*,  then  equation  (23)  becomes 

d2u  fi  d-u       I         fx\ 

Accordingly,  the  equation  of  the  orbit  depends  on  the  sign  of  1  -—^  und 
therefore  on  the  initial  circumstances  of  the  motion. 

Suppose  the  particle  projected  initially  at  the  distance  R,  with  the  velo- 
city V,  and  in  a  direction  which  makes  the  angle  w  with  R  ;  then 

h  =  VR  sin  w. 
Again,  if  V  be  the  velocity  in  the  equidistant  circle,  we  have,  Art.  89, 

j?2'      '"A2       F2i£3  sin2  w       \Fsm«/ 


200  Central  Orbits. 

Hence  1  —  —  is  positive,  zero,  or  negative,  according  as  V  sin  w  is  greater, 
equal  to,  or  less  than  V',  the  velocity  in  an  equidistant  circle. 

IX    . 

(1)  Let  Fsin  a>  >  V.    In  this  case  1  —  —  is  positive — equal  A"2,  suppose — and 


the  equation  may  be  written 

d2u 


A" 


The  integral  of  this  is  of  the  form 

u  =  A  cos  (k9  +  a). 

A  is  plainly  the  maximum  value  of  u ;  and  therefore  corresponds  to  an  apsidal 
distance.  Let  a  be  this  distance,  and,  if  6  be  measured  from  the  apsidal  line, 
the  equation  of  the  orbit  is 

r  cos  kd  =  a.  (1) 

M 

(2)  Let  V  sin  a>  =  V\  then  I  -  —  =  0,  and  we  have 

—  -0- 

d0~  ~     ' 

this  gives  u  =  A  (0  +  a) ;  and  the  equation  of  the  orbit  is  reducible  to 

r0  —  constant,  (2) 

which  represents  the  hyperbolic  spiral. 

(3)  Let  V  sin  <o  <  V .     If  we  multiply  the  equation 

d'lu  ix 

^  +  B  =  pM 

by  2du,  and  integrate,  we  get 

where  c  is  constant. 

Hence  v2  =  fxu1  +  h2e. 

Substituting  the  initial  values,  this  gives 
yi.  _  yn 

du\2    m    a  m    v* -  r,z 


therefore  ( -^ )  +  «2  =  ^ -u2  -t 

\dU/  br 


h2 


(3) 


The  apse  is  determined  by  making  —  =  0  ;  consequently,  since  —  >  1,  the 

d9  h* 

orbit  has  or  has  not  an  apse  according  as  V  is  less  or  greater  than  V . 

Hence,  if  the  initial  velocity  be  less  than  that  in  an  equidistant  circle,  the 
orbit  is  apsidal. 


Examples.  201 


Suppose  a  to  be  the  corresponding  apsidal  distance,  then 
F'2  -  F2 


=MM 


and,  making  ^  —  1  =  k-,  equation  (3)  becomes 

\dd)       a2  v  ' 

therefore  — -  =  -  V  a~n2  -  1  ;    or  =  Arf0. 

d0      0  Va2  w2  -  1 

The  integral  of  this  is  


he  +  a  =  log  («w  +  V«-w2  -  1). 

But  if  a  be  measured  from  the  apse,  -we  have  0  =  0  when  au  -  1.     Conse- 
quently a  =  0,  and  we  have 

au  +  V«2m'-  -  1  =  <^- 
Hence  ««  =  |(^  + «"**). 

Here  u  increases  with  6  ;  and  consequently  the  body,  after  leaving  the  apse, 
approaches  nearer  and  nearer  to  the  centre  of  force. 

Secondly,  if  the  initial  velocity  be  equal  to  that  in  the  equidistant  circle, 
(3)  becomes 

du-  du 

— —  =  &2«2,  or  — -  =  ku ; 

dd2  dd 

this  gives  u  =  aeke, 

the  equiangular  spiral  (Ex.  7). 

Thirdly,  if  V  be  greater  than  F',  let 


and  equation  (3)  becomes 


=  *2J82, 


This,  when  integrated  as  above,  gives 

w  +  Vw2  +  £2  =  -4***, 
and  the  curve  is  represented  by  the  equation 

2«  =  ^  _  ^'  r* 

A 

The  value  of  -4  can  be  readily  determined  from  the  initial  conditions. 

17.  In  elliptic  motion  about  a  centre  of  force  in  a  focus,  prove  that  Jvcfo, 
taken  through  any  arc,  is  proportional  to  the  area  subtended  by  the  arc  at  the 
empty  focus. 


202  Central  Orbits. 

18.  Prove  that  the  expression  for  the  central  attraction  for  any  law  of  force 
may  be  written  in  the  form 

F=  —  -r. 

r3 

If  we  change  the  sign  in  the  expression  for  the  acceleration  along  the  radius 

vector  in  Art.  28,  we  get 

F=rO°~-r. 

h 
This  assumes  the  proposed  form  on  substituting  for  6  its  value  -^ . 

19.  "What  would  be  the  motion  of  a  projectile  if  the  force  of  gravity  varied 
inversely  as  the  cube  of  the  height  above  a  horizontal  plane  ? 

Here  the  path  evidently  lies  in  a  vertical  plane. 

If  the  line  of  intersection  of  this  plane  with  the  horizontal  plane  be  taken 
as  the  axis  of  x,  and  a  vertical  line  as  the  axis  of  y}  the  equations  of  motion 
may  be  written 

dt2         '    dfi  v/3' 


therefore  — =«,    ^-_  +  /, 

where  e  and  c'  are  constants  which  depend  on  the  initial  circumstances  of  the 
motion.     Consequently 

'dy\  2      1  jx  +  c y2 


(dyy=  I 
\dx)       c* 


ydy  dx 


VV  +  c'  y'1        c 


Hence  we  get  V/j.  +  c'y'2  =  c'  -  +  const. 


c 


Consequently  the  path  is  an  ellipse  or  a  hyperbola  according  as  c'  is  negative 
or  positive.     The  path  is  a  parabola  if  c'  —  0. 

20.  Prove  by  Newtonian  methods  that,  if  two  bodies  attract  one  another 
according  to  any  law,  they  describe  similar  figures  about  their  centre  of  inertia 
and  about  one  another. 

Neglecting  the  obliquity  of  the  ecliptic,  and  the  inclination  and  the  eccen- 
tricity of  the  lunar  orbit,  show  that,  if  we  take  the  Sun's  distance  as  390  times 
that  of  the  Moon,  the  Earth's  mass  as  79  times  that  of  the  Moon,  and  the  lunar 
synodic  period  as  30  mean  solar  days  ;  then  the  solar  day  is,  to  a  near  approxi- 
mation, shorter  at  full  Moon  that  at  new  Moon  by  one  468,000th  part  of  a  mean 
solar  day.  '  Comb.  Trip.,  1882. 

21.  A  material  particle,  moving  freely  in  a  plane,  being  supposed  to  describe 
a  conic  under  the  action  of  a  central  force  emanating  from  any  point  in  the 
plane  ;  show  that  the  force  varies  directly  as  the  distance  from  the  point,  and 
inversely  as  the  cube  of  its  distance  from  the  polar  of  the  point  with  respect  to 
the  curve. 


Examples.  203 

22.  In  free  motion  in  a  plane  under  the  action  of  a  central  force  varying 
according  to  any  law,  state  and  prove  the  effect  on  the  trajectory  (and  on  the 
motion  in  it)  of  an  additional  force  emanating  from  the  same  centre,  and  varying 
inversely  as  the  cuhe  of  the  distance. 

23.  An  ellipse  of  eccentricity  e  and  a  parabola  have  a  common  focus  and 
latus  rectum  ;  and  equal  particles  describe  them  under  the  action  of  forces,  to 
the  common  focus,  of  the  same  absolute  intensity.  If  the  particles  moving  in 
the  same  direction  meet  at  one  extremity  of  the  common  latus  rectum  and  coa- 
lesce, prove  that  their  subsequent  path  will  be  an  ellipse  of  eccentricity  §-(1  ±e), 
according  as  both  foci  of  the  ellipse  do  or  do  not  He  within  the  parabola ;  and 
find  its  major  axis.  What  will  the  path  be,  if  the  particles  be  moving  in  oppo- 
site directions  when  they  meet  ?  Gamb.  Trip.,  1879. 

24.  A  body  is  revolving  in  an  ellipse,  whose  eccentricity  is  >  \,  under  the 
action  of  a  force  tending  to  the  focus  S;  and  when  it  is  at  a  distance  SP  from  S 
equal  to  the  latus  i*ectum,  a  blow  is  given  to  it  perpendicular  to  SP,  such  that 
its  new  direction  is  perpendicular  to  the  major  axis.  Show  that  the  dimensions 
of  the  orbit  are  unaltered,  but  that  the  major  axis  is  turned  through  an  angle 
SPS,  where  R  is  the  empty  focus.  Id.,  1882. 

25.  Find  the  laws  of  attraction  for  which  the  trajectories  described  round  a 
centre  of  force  are  closed  orbits.  (Bertrand,  Comptes  rendus,  1873.) 


**m 


F(u)  +  const.,  equation  (23)  gives 


o       lduY~      1    „,  , 


where  c  is  an  arbitrary  constant ;  therefore 

du 


-  F(u)  +e-tfi 


Again,  let  a,  /3  represent  the  values  of  u  which  correspond  to  the  apsidal 
distances,  then  a  and  £  are  roots  of  the  equation 

^  F{u)-l  c-n*=0. 

Accordingly  we  must  have 

i*(a)  +  c-a*=0,     i-F(j8)+*-/3*  =  0; 

and  if  0O  be  the  apsidal  angle,  we  get,  abstraction  being  made  of  the  sign, 
,.£  du 

00 


}-rF(u)  +  c-ui 

>a\J  A- 

Assuming  >n9o  =  7r,  then,  for  a  closed  orbit,  m  must  be  a  commensurable  number 
(Art.  184). 


204  Central  Orbits. 

If  —  and  c  be  eliminated  by  aid  of  the  two  preceding  equations,  we  obtain 

du  TV 


w 


,*»-*'(«)       9F(u)-F(») 
P  F(fS)-F(a)      °   F(/3)-F(a) 


an  equation  which  should  hold  for  all  values  of  o  and  £. 

To  determine  the  form  of  the  function  F,  we  suppose  o  and  #  very  nearly 
equal,  in  which  case  the  orbit  is  approximately  circular. 

Hence,  from  Art.  187,  we  get 


J%  I  1        aF"{a)      or    aF"{a)       1 


Let  F'(a)  =  F,  then  ^T  =  ^^  =  (i  -  nfi)  — ;  hence 
p  .r  (a)  a 

where  C  is  an  arbitrary  constant.  (d) 

From  this  we  get  F(a)  = ■.  o2-"'2  +  const. 

°  2  -  w2 

We  may  assume  the  latter  constant  to  be  zero,  since  it  disappears  when  we 
substitute  in  equation  (c). 

Again,  since     2/t  ^  =  F'(u),  we  have  £(«)  =  fl  m3-"'3, 

where  Ci  is  arbitrary. 

"We  next  proceed  to  determine  m  from  the  condition  that  (c)  must  be  satis- 
fied for  all  values  of  o  and  £. 

(1)  Let  ?n2  <  2,  and  make  o  =  0,  and  /3  =  1  ;  then 

F(.)  =  0,    flfl-j^ 5- 

Substituting  in  (c),  we  obtain 

dw  IT 


i: 


Jo  VV-™    -  W2 

Again,  if  u™*  =  z, 


Jo  wv^^ri "  ™2  Jo  v^r^) 


The  condition  gives  —  =  — -  :  therefore   m  —  1.      Accordingly,  the  forc< 
mm2 

1 
varies  as  «*,  or  as  — -  • 
r2 


Examples.  205 

(2)  Let   or  >  2  ;  then,  if  o  =  0,  we  have  F(a)  =  F(0)  =  -  oo  ;  and  if  j3  =  1 
(y 

we  have  F(&)  =  F(l)  = .     Substitute  in  (c),  and  it  becomes 

1  —  w 

l'1  dti  IT  IT  IT 

1  -  =  —  ;    or  —  =  —  ;     ,\  m  =  2, 

Jo  V 1  —  ur      m  2      m 

in  which  case  the  force  varies  directly  as  the  distance. 

Hence,  as  M.  Bertrand  observes,  "parmi  les  lois  d'attraction  qui  supposent 
Taction  nulle  a.  une  distance  infinie,  cella  de  la  nature  est  la  seule  pour  laquelle 
un  mobile  lance  arbitrairemoit,  avec  une  vitesse  inferieure  a  une  certaine  limite, 
et  attire  vers  un  centre  fixe,  decrive  neoessairement  auteur  de  ce  centre  une 
courbe  fermee.  Toutes  les  lois  d'attraction  permettent  des  orbites  fermees,  mais 
la  loi  de  la  nature  est  la  seule  qui  les  impose.'''' 

26.  Investigate  the  condition  of  stability  of  a  circular  orbit  described  about 
a  centre  of  attraction  in  the  centre  of  the  circle. 

Prove  that  if  the  attraction  varies  inversely  as  the  fourth  power  of  the  dis- 
tance, a  particle  describing  a  circle  of  radius  a  freely  will  be  found  ultimately 
describing  either  the  curve 

cosh  e  +  1  cosh  0-1 

>•  =  a  — ■ ,  or  r  =  a  — : • 

cosh  e  -  2  cosh  0  +  2 


206 


CHAPTEE  VIII. 

CONSTRAINED    MOTION — MOTION    IN    A    RESISTING    MEDIUM. 

Section  I. —  Constrained  Motion. 

189.  Motion  on  a  Fixed  Curve. — When  a  particle  is 
constrained  to  move,  without  friction,  on  a  given  fixed  curve, 
the  problem  reduces  to  the  determination  of  the  velocity  at 
any  instant,  as  well  as  of  the  normal  reaction  of  the  curve. 
The  motion  may  in  this  case  be  regarded  as  free  by  the 
introduction  of  the  force  of  reaction  of  the  curve,  in  addition 
to  the  external  forces. 

Hence,  if  JV  represents  the  normal  reaction,  the  general 
equations  of  motion  may  be  written,  when  referred  to  a  rect- 
angular system  of  axes, 

m-r^  =  X  +  Ncosa,  m  -^  =  Y+N cos/3,  m  — ■=■  =  Z  +  N  cos  7, 
dt'  dt'  dt 

(i) 

where  a,  /3,  7  are  the  angles  the  normal  reaction  makes  with 
the  axes  of  coordinates  ;  and  X,  Y,  Z  are  the  components  of 
the  external  force,  parallel  to  the  axes  of  coordinates,  respec- 
tively. If  the  first  equation  be  multiplied  by  dx,  the  second 
by  dy,  and  the  third  by  dz,  we  get,  on  addition, 

"'  (J? * + S rfy + 8  * )  ~ x<h + Yibj + zdz'  (2) 

since  cos  adx  +  cos  fidy  +  cos  7  dz  =  0,  as  the  direction  of  N  is 
perpendicular  to  the  tangent  to  the  curve. 
This  gives  on  integration 


\  mv2  =  %  m 


1™  \(dx\\(dl\\(dz 


(IMfHs)i 


{Xdx  +  Ydy  +  Zdz) 
+  const.      (3) 


Motion  on  a  Fixed  Curve.  207 

Hence  the  velocity  is  given  by  the  same  equation  as  in 
the  case  of  unconstrained  motion  (Art.  131). 

For  a  conservative  system  of  forces  (Art.  125),  the  velo- 
city v  at  any  point  can  generally  be  found  from  this  equation. 
For,  let  Xdx  +  Ydy  +  Zdz  be  the  exact  differential  of  the 
function  <j>  (%,  y,  z)  ;  then  if  v  be  the  velocity  at  the  point 
,/,  y\  zy  we  have 

\m  {f  -  v2)  =  <p  (x,  y,  %)  -  <j>  (of,  y\  z).  (4) 

Hence  the  velocity  at  any  point  is  independent  of  the  path 
described ;  and,  accordingly,  if  different  curves  be  drawn 
joining  any  two  points,  a  particle  starting  from  one  of  these 
points  with  a  given  velocity  would  arrive  at  the  other  point 
with  the  same  velocity  whatever  path  it  described ;  friction 
being  neglected. 

Two  of  the  preceding  equations  (1)  are  sufficient  for  a 
plane  curve  ;  for  in  this  case  N  acts  in  the  plane  of  the 
curve,  and,  by  taking  the  axes  of  x  and  y  in  that  plane,  the 
third  equation  will  disappear. 

In  the  case  of  a  central  force,  represented  by  ju<£'(r)>  we 
have,  as  in  Art.  131,  *      . 

i«(*-O  =  M(*(r)-#(r0). 

Again,  as  in  Art.  116,  it  is  readily  seen  that  the  pressure 
on  the  curve  in  any  case  is  the  resultant  of  the  centrifugal 
force  and  the  normal  component  of  the  external  forces. 

The  particle  will  leave  the  curve  at  the  point  for  which 
the  normal  reaction  becomes  zero. 

Examples. 

1.  A  particle  is  constrained  to  move  in  a  circle  under  the  influence  of  a  re- 
pulsive force,   acting  from  a  point  on  the  circumference,  and  varying  as  the     ^ 
distance  :   find  the  pressure  on  the  curve,  the  initial  position  being  at  the  centre 

of  force,  and  the  particle  starting  from  a  state  of  rest. 

j£nSt     J^L  }  where  r  is  the  distance  from  the  centre  of  force,  and  a  the  radius 
%a  of  the  circle.    I 

2.  A  particle  is  constrained  to  move  in  a  logarithmic  spiral,  and  is  attracted 
to  the  pole  of  the  spiral  by  a  force  varying  inversely  as  the  square  of  the  dis- 
tance. If  the  particle  start  from  rest  at  the  distance  a  from  the  pole,  find  the 
time  of  describing  any  portion  of  the  curve. 


208  Constrained  Motion, 

Let  ix  denote  the  absolute  force  ;  then,  by  (5),  we  have 


ds      rr      x      1 


Again,  if  r  =  ce*  be  the  equation  of  the  spiral,  we  have 
ds      dr    , -, 


therefore  —  ==    /  — —   A 

dt      \  1  +  k~  \  r      a 

Integrating,  as  in  Art.  140,  we  get  for  the  time  of  motion  from  the  distance 
a  to  the  distance  r, 


-J*£fi(.-j2+v^) 


Also  the  whole  time  of  motion  to  the  centre  is  -  A  /  — • 

2  \  2yU 

It  is  readily  seen  that  the  problem  of  constrained  motion  in  a  logarithmic 
spiral,  under  the  action  of  any  central  force  directed  to  its  pole,  is  reducible  to 
free  rectilinear  motion  under  the  action  of  a  corresponding  central  force  in  the 
line  of  motion. 

3.  A  particle  under  the  action  of  gravity  moves  down  the  inner  side  of  a 
smooth  ellipse  whose  axis  major  is  vertical.  Being  given  its  initial  velocity, 
find  where  it  will  leave  the  ellipse. 

Taking  the  centre  as  origin,  and  the  axis  major  as  axis  of  x,  the  value  of  x 
at  the  required  point  is  given  by  the  equation 

2d  =  Zx  -  &  -4, 

where  d  is  the  height  above  the  centre  of  the  level  line  to  which  the  velocity  at 
each  point  is  due. 

4.  In  the  same  question  find  the  least  velocity  at  the  lowest  point  of  the 
ellipse  in  order  that  the  particle  should  make  a  complete  revolution  in  the  curve. 

Am.    \/(/a  (5  —  e2). 

190.   Theorem    of  M.   Ossian    Bonnet. — If    masses 

m,  m',  m",  &c,  respectively  subject  to  the  action  of  forces, 
F,  F\  F'\  &c,  and  starting  all  in  the  same  direction  from  a 
point  A,  with  velocities  t?0,  v0',  v",  &c,  describe  the  same  curve 


Theorem  of  M.  Ossian  Bonnet.  209 

ACB ;  then  the  same  path  will  also  be  described  by  the  mass 
M,  when  projected  from  the  same  point  in  the  so;me  direction, 
and  subject  to  the  action  of  all  the  forces,  F>  F',  F",  &c, 
provided  the  initial  vis  viva  MV02  is  equal  to       c 

mv02  +  m'i\?  +  m"v"%  +  &c, 

the  sum  of  the  vires  vivce  of  the  different  masses.  (Bonnet, 
IAouville's  Journal,  1844.) 

For,  suppose  the  particle  M  constrained  to  mov^e  in  the 
curve  ACB,  and  let  iVbe  the  normal  reaction  at  any  point ; 
then,  if  the  components  of  F,  parallel  to  a  rectangular  system 
of  axes,  be  respectively  represented  by  X,  F,  Z,  those  of  F\ 
by  X\  Y\  Z\  &c. ;  from  (1),  we  have 

M-£  =  X  +  X'  +  X"  +  &c.  +  i^cos  a. 
dt~ 

M^2  =  Y  +  Y'  +  Y"  +  &e.  +  iV^cos  j3, 

M^  =  Z  +  Z'  +  Z"  +  &o.  +  i^cos  7, 
at" 

and,  as  in  (2),  we  have 

d(MV2)  =  2dx2X  +  2dy^Y+2dz^Z. 

But  if  v,  v\  v'\  &c,  be  the  velocities  in  the  partial  movements 
of  m,  m',  m\  &c,  at  the  same  point, 

d(mv2)  =  2  (Xdx  +  Ydy  +  Zdz), 

&c,  &c,  &c. 

Hence     d (MV2)  =  d (mv2  +  mvn  +  m'v"z  +  &c.) ; 

therefore  MV"  =  2(m#2)  +  constant, 

or  M  V2  =  2w#2,  from  our  hypothesis. 

It  is  now  easy  to  prove  that  the  normal  pressure  iV  is  zero 
at  each  point,  and  consequently  that  M  would  describe  the 
curve  ACB  freely,  under  the  combined  action  of  all  the 
forces. 

p 


210  Constrained  Motion. 

For  the  force  N  is  equal  and  opposite  to  the  resultant  of 

MV2 

the  centrifugal  force,  ,  and  the  several  normal  compo- 
nents of  the  forces,  F,  F\  F" ,  &c. 

Again =  —  + + +  &c. ;  (o) 

P  P  P  P 

but  — ,  - — ,  &c.j  are  respectively  equal  and  opposite  to  the 

P   -    P 
normal   components  of  F,  F\  F"9  &c,  because  m,  m\  &c, 

describe  the  path  A  CB  freely. 

Hence  there  is  equilibrium  between  the  centrifugal  force 

MV2 

and  the  total  normal  component  of  F,  F\  Fr\  &c. ;  and 

P 

consequently  N  =  0. 

In  general,  if  the  initial  velocity  of  M  do  not  satisfy  the 
equation  MV02  =  ^nv02,  the  normal  pressure  on  the  path  ACB 
mil  vary  directly  as  the  curvature.  For,  from  the  preceding 
analysis, 

jsr= = .  (6) 

p  p 

Also,  if  one  of  the  forces  (Ff  suppose)  be  changed  into 
its  opposite,  it  is  readily  seen  that  the  preceding  theorem  still 
holds,  provided  we  change  the  sign  of  the  corresponding  term 
(mV2)  in  the  expression  S(m©2). 

Examples. 

1 .  A  particle  constrained  to  move  in  an  ellipse  is  acted  on  by  an  attractive 
force  directed  to  one  focus,  and  a  repulsive  force  from  the  other,  whose  intensi- 
ties vary  as  the  inverse  square  of  the  distance  :  if  the  absolute  intensities  of  the 
forces  be  equal,  find  the  pressure  on  the  ellipse  at  any  point  during  the  motion. 

2.  Hence  show  that  a  particle  placed  at  equal  distances  from  two  such  centres 
of  force  will  describe  a  semi-ellipse,  under  their  joint  action. 

3.  A  particle  moves  under  the  attraction  of  two  forces  directed  to  the  fixed 
points  A  and  B,  each  varying  according  to  the  law  of  nature,  and  a  third  force, 
varying  directly  as  the  distance,  directed  to  C,  the  middle  point  of  AB  ;  show 
that  the  particle  can  be  projected  from  any  point  so  as  to  describe  an  ellipse 
having  A  and  B  as  its  foci.  Lagrange,  Mec.  Anal.,  t.  2,  §  83. 

Ans.  The  initial  velocity  v0  is  given  by  the  equation 

t'o2  =  —z  +  -j,  +  /*  J  J  > 


Motion  on  a  Fixed  Surface.  211 

where  /x,  /x',  ll"  denote  the  ahsolute  forces  for  the  centres  A,B,  C,  respectively  ; 
/,/'  the  initial  distances  from  A  and  B  ;  and  a  the  semiaxis  major  of  the  ellipse. 
The  initial  direction  of  motion  must  bisect  the  external  angle  formed  by  the 
lines  joining  A  and  B  to  the  point  of  projection. 

4.  In  the  same  case,  if  the  particle  he  constrained  to  move  in  the  ellipse, 
find  the  reaction  R  at  any  point  during  the  motion. 

Am.    Rp  =  m  (~-  -f  ^-  +  n"ff  -  tv  j  , 

where  p  is  the  radius  of  curvature  at  the  point. 

5.  If  a  material  particle,  moving  freely  under  the  action  of  gravity,  he  dis- 
turbed by  the  action  of  a  central  force  varying  inversely  as  the  square  of  the 
distance  ;  determine  the  circumstances  of  its  projection  from  a  given  point,  in 
order  that  it  may  describe  a  parabola  in  a  vertical  plane  having  its  focus  at  the 
centre  of  force. 

191.  Motion  on  a  Fixed  Surface. — If  a  particle  be 
constrained  to  move  on  a  smooth  surface,  the  general  equa- 
tions of  motion  are  plainly,  as  in  (1), 

d2x     _     „  (Pu     _     ,T        _.        d2z      _     __ 

w—  =  X+Jy  cos  a,   m  —  =  F  +  iV  cosp,  m  —  =  Z  +  N cos  y, 

C(l~  CIL  1(0 

where  a,  /3,  y  ar©  the  direction  angles  of  the  normal  to  the 
surface. 

It  is  obvious  that  in  this  case  also  the  velocity  at  any 
point  is  determined  by  the  equation 

j>mv2  =  J  [Xdx  +  Ydy  +  Zdz)  +  const.  (7) 

If  gravity  be  the  sole  acting  force,  and  the  axis  of  z  be 
taken  in  the  vertical  direction,  our  equations  may  be  written 

d  ~x  d^u  d~z 

— r  ^i^cosa,   -j~  =  Ncos(3,   —  =  Ncosy  -  g.       (8) 

c(t"  (it"  (It 

When  the  surface  is  one  of  revolution  round  a  vertical 

axis,  the  normal  at  each  point  intersects  that  axis  ;  and  if  n 

denote  its  length,  we  have 

x  „       II 

cos  a  =  - ,    cos  p  =  -  • 
n  n 

Hence  the  two  former  equations  give 

dhj        d'x      . 

P  2 


212  Constrain  ed  Mo tion . 

or,  on  integration, 

dy        dx 

where  c  is  a  constant. 

This  equation  shows  that  the  point  of  projection  on  a 
horizontal  plane  describes  equal  areas  in  equal  times  round 
the  point  in  which  the  axis  of  revolution  meets  the  plane. 

192.  Motion  on  a  Spherical  Surface.— We  shall 
apply  what  precedes  to  the  motion  of  a  particle  under  the 
action  of  gravity  on  a  smooth  sphere.  This  contains  the 
general  question  of  the  motion  of  a  simple  pendulum,  and  is 
called  the  problem  of  the  spherical  pendulum.  Taking  the 
centre  as  origin,  and  the  positive  direction  of  the  axis  of  z 
downwards,  the  equation  of  the  sphere  is 

x2  +  y2  +  z2  =  a2, 
where  a  is  the  radius. 

Also  the  general  equations  of  motion  may  be  written 

x=N-,  y  =  N-,    z  =  N-  +  g, 
a  a  a 

adopting  Newton's  notation  (Art.  23). 

From  the  first  two  equations  we  get,  as  before, 

xij  -yx  =  c.  (9) 

Also,  as  in  (7), 

x2  +  y2+  z2  =  V,?+2g(z-a), 

where  V0  represents  the  velocity  corresponding  to  z  =  a. 
Again,  differentiating  the  equation  of  the  sphere, 

xx  +  yij  +  zz  =  0,    or    xx  +  yy  =  -  zz. 

If  this  be  squared  and  added  to  (9),  when  also  squared, 
we  get 

{x2  +  y2)  (x2  +  y2)  =c2  +  z2z2. 


Motion  on  a  Spherical  Surface.  213 

Hence         (a2  -  z2)  {  V02  +  2g  {z  -  a)  -  z2  j  =  c2  +  z2 z2, 

or  a2z2  =  (a2  -  z2)  {  V02  +  2g(z-a))- c\  (10) 

The  subsequent  investigation  is  simplified  by  supposing  V0  to 
correspond  to  the  lowest  point  in  the  path  of  the  particle  ;  for, 
since  the  motion  at  that  point  is  horizontal,  we  have  z  =  0 
when  z  =  a,  and  consequently 

c2  =  {a2-a2)Vo2  =  2gh{a2-a2), 

if  h  be  the  height  to  which  the  velocity  V0  is  due. 
Substituting  this  value  for  c2  in  (10),  we  get 

// 

a2z2  =  2g{a-z)\z2  +  h(z  +  a)-a2}. 

Again,  the  expression  z2  +  h  (z  +  a)  -  a2  may  be  written 

(z-f5){z  +  y),  where 

a2-fi2  a2  +  afi 

h   m    __       and   y    .   __.  (11) 

Accordingly 

therefore     a'z  =  a  —  =  -  s/2g{a  -  z)  (z  -  [5)  {z  +  y).  (12) 

a  v 

The  negative  sign  must  be  taken  since  %  diminishes  with  t, 
which  is  reckoned  from  the  instant  the  particle  is  in  its 
lowest  position. 

Also,  when  s=(3we  have  z  =  0,  and  the  motion  is  again 
horizontal.  It  is  readily  seen  that  during  the  motion  z  must 
lie  between  the  limits  a  and  /3  ;  and  consequently  the  path 
of  the  particle  is  a  tortuous  curve  lying  between  two  horizon- 
tal lesser  circles  on  the  sphere ;  we  accordingly  may  assume 

z  =  a  cos2<£  +  (3  sm2(j),  (13) 

and,  substituting  in  (12),  get 


2a  -j~  =  */2g  (a  cos"^  +  j3  sin'^  +  y), 
ao 


214  Constrained  Motion. 

Hence,  since  t  =  0  when  <p  =  0,  we  have 


where  k~  = 


sin"0 

a  -  j3  a2  -  j32 


a  +  7       cr  +  2aj3  +  a2 

Consequently  the  time  of  motion  depends  on  an  elliptic 
function,  and  is  reducible  to  that  of  the  description  of  a  cor- 
responding arc  in  a  vertical  circle  (Arts.  101,  114). 

Again,  if  T  denote  the  period  of  a  vibration,  that  is,  the  time 
of  motion  from  a  lowest  to  a  consecutive  lowest  position,  we 
have 


-*-  p;      r* 

>a|3+a2)Jo 


V^(«2  +  2a/3+a2)Jo  ^/i  _/^sin20 

a2  -  a2 

It  may  be  observed  that  when  a  =  ]3,  we  have  A  =  — ~ — , 

and  the  question  reduces  to  that  of  the  conical  pendulum, 
already  considered  in  Art.  112. 

Next  let  \p  be  the  angle  that  the  vertical  plane,  passing 
through  the  centre  and  the  position  of  the  particle  at  any 
instant,  makes  with  the  plane  of  z%,  then  y  =  x  tan  \p ;  and 
consequently 

dlJ     „dx     ^  d  (V\_  ^_2.,  dxP 


c  =  x-tt  -  y— =  a?2—   -  =#2sec2i/>  , 
dt      J dt        dt\x  r dt 


«+/)f-(^--D«.     (15) 


Also     c=yw^?)=  h^-aw-m.     (16) 

\  a  +  p 

a  W~* 
\  a 


/3 

'^> .  _  ^  •(...)(,.i3)(,  +  7,  «, 

and  the  angle  <//  is  represented  by  an  elliptic  function  of  the 
third  species,  thus 


{a2  -  a2)(a2  -  /32)  f -JL 


*-%/ ^ 1  (a 


52)v/(a-s)(s-/3)(i3  +  7) 


(17) 


Small  Oscillations.  215 

In  the  projection  of  the  path  on  the  horizontal  plane 
through  the  centre,  the  greatest  and  least  distances  from  the 
centre  correspond  to  the  greatest  and  least  values  of  s,  i.  e.  to 
z  =  a  and  z  =  /3.  These  are  called  the  apsidal  distances,  and 
the  corresponding  angle,  the  apsidal  angle  of  the  path.  If  ¥ 
be  the  apsidal  angle  its  value  is  represented  by  the  integral 


dz 


|(*-«»)(*-P')p «z  (18) 

193.  Small  Oscillations.— If  the  particle  make  a  small 
oscillatory  motion  round  the  lowest  point,  we  may,  as  a  first 
approximation,  make  a  =  a,  j3  =  a  in  (14).     This  gives 

7v  =  0,  and*=tf>   /?.  (19) 

Next,  if  z  =  a  cos  0,  a  =  a  cos  0O,  j3  =  «  cos  0i,  the  equa- 
tion 

z  =  a  cos2<£  +  /3  sin20 

gives  02  =  0o2cos2<£  +  012sin2<£,  neglecting  powers  of  0,  0O,  0t 
beyond  the  second. 

Also  (16)  gives  in  this  case 

\}a 
.-.  by  (15),  we  have 

dt  =''    02  \a      0o2oos80  +  0i8sinV\a 

Consequently,  by  (19), 

#  _  0Q0i   

<:/<£       0O2  cos2<£  +  0i2  smV 

Hence,  by  integration, 

.      0i. 
tan  \p  =  7j-  tan  0, 

"o 


216  Constrained  Motion. 


sin  \L      0,  sin  <f> 
or  -f  = r  ; 

COS  \p        0O  COS  (f) 

hence  we  have 

.     ,      0i  sin  0 
Sm  *  =  — 0      ' 

,  0O  COS  0 

and  cos  ^  =  — ^ — . 

u 

Moreover,  to  the  same  degree  of  approximation,  we  have 

x  =  a  9  cos  \p,     y  =  a  9  sin  \f, ; 
accordingly, 

#  =  «  0O  cos  $,     y  =  a9i  sin  0 ; 

•••  g?+Jr«-  (20) 

This   shows  that  the   horizontal  projection  of  the  path  is, 
approximately,  an  ellipse,  whose  semiaxes  are  a90  and  adi. 

The  next  approximation  is  given  in  the  following 
examples. 

The  general  problem  of  the  spherical  pendulum  appears 
to  have  been  first  fully  discussed  by  Lagrange :  see  Nee. 
Anal.,  t.  2,  sect.  8. 


Examples. 

1.  If  a  particle  perform  small  oscillations  about  the  lowest  point  on  a  sphere, 
investigate  its  motion  to  an  approximation  of  the  second  order. 

It  is  here  more  convenient  to  transfer  the  origin  to  the  lowest  point  on  the 
sphere,  and  to  take  the  positive  direction  of  z  upwards.  Accordingly,  we  sub- 
stitute   z  =  a-z',     a  =  a-a,     0  =  a-/3',     when  equation  (12)  becomes 


Examples, 

Hence,  removing  the  accents,  we  get 

dz 


217 


\ff)a 


Vfr-w-^-s+ssfefl)' 


where  £  and  o  represent  the  distances  of  the  highest  and  lowest  points  in  the 
path  from  the  plane  of  xy. 

Again,  if  a  and  /3  be  both  so  small  that  their  higher  powers  may  be  neglected, 
we  obtain 

dz 


V'J.^-«)o-,)(i-^) 

2\9„a<J[z-a)US-z)      *y/ga)a 


zdz 


(*- a)*  (£-«)* 


neglecting  the  subsequent  terms,  since  —  is  a  very  small  fraction. 
Hence,  if  z  =  o  cos2<£  +  /3  sin2<£,  we  get 

^_     /"</>+ —I    (a  cos20  + /3  sin2^)  cfy> 

\y  Ay/agJo 

Consequently,  if  T  be  the  whole  time  of  motion  from  one  lowest  position  to  a 
consecutive  one,  we  get 


'•'£(l  +  '-& 


Again,  to  find  the  apsidal  angle  to  the  same  degree  of  approximation. 
Transforming  the  origin  in  equation  (16)  to  the  lowest  position,  we  readily 
obtain 


*  =  "Vaj8 


dz 


z(2 


--■^-oM'-i&^S) 


218  Constrained  Motion. 

Hence,  since  as  before  we  may  take 


we  get 


2« 

-a-j8 

1 

(2a- 

a) (2a  - 
dz 

0) 

2a' 

if?  =  «S  V2aj3 

Ja  (2a-«)SaV(z-  a)  (j8  -  z) 


J«iV(«-o)0-«) 

J««V(*-a)(fl-«)       8    a     J„ 


,V(«-«)0B-») 

neglecting  the  subsequent  terms  as  before. 
Substituting  o  cos2</>  +  j3  sin2<£  for  z,  we  obtain 

3Voj8 


,  /    1 0  \       3  Vaj3 

tan-^-taa^+j— *. 


Hence,  taking  <p  between  the  limits  0  and  -,  tbe  apsidal  angle  is  given,  ap- 
proximately, by  tbe  equation 

3  v'ojS 


v  /,      3  vo)8\ 


This  sbows  tbat  in  tbe  approximate  elliptic  path  tbe  apse  continually  progresses. 
Again,  if  p,  q  denote  the  small  apsidal  distances,  or  the  semidiameters  of  tbe 
approximate  elliptic  path,  we  get 


2  \    t8«s/ 


Accordingly,  the  rate  of  progression  of  the  apse  varies  approximately  as  the 
area  of  the  projection  of  tbe  path. 

2.  Prove  that  tbe  pressure  on  the  sphere  is  given  by  the  equation 

a     {  o  +  /3 

3.  If  a  particle  be  projected  with  a  given  velocity  along  the  horizontal  great 
circle  of  a  smooth  hollow  sphere,  find  at  what  point  its  vertical  velocity  will  be 
greatest. 

Ans.    z  = ,  h  being  tbe  height  due  to  the  velocity  of  projection. 


Rectilinear  Motion  in  a  Residing  Medium.  219 

4.  A  particle  is  projected  horizontally  along  the  interior  surface  of  a  fixed         V 
smooth  hemisphere,  the  axis  of  which  is  vertical,  and  vertex  downwards.    Given 

the  point  of  projection,  determine  the  velocity  so  that  the  particle  may  ascend 
exactly  to  the  rim  of  the  hemisphere.  iTjjI 

Atis.     a    /— . 
\  a 

5.  If  a  particle  move  on  the  interior  surface  of  a  paraboloid  of  revolution, 
whose  axis  is  vertical,  prove  that  the  velocity  at  the  highest  point  in  the  path 
is  that  due  to  the  height  of  the  lowest  point  above  the  vertex  of  the  paraboloid ; 
and  similarly  for  the  velocity  at  the  lowest  point. 

6.  In  the  last  question  show  that  the  pressure  at  any  point  P  varies  as  the 
curvature  of  the  meridian  at  that  point ;  and  that  the  resolved  vertical  pressure 
is  to  the  weight  of  the  particle  as  SL  x  SM\  SP2,  where  L  and  il/"are  the  highest 
and  lowest  points  of  the  path,  and  S  the  focus. 

Section  II. — Rectilinear  Motion  in  a  Resisting  Medium. 

194.  If  a  mass  m  be  supposed  to  move  in  a  straight  line, 
without  rotation,  in  a  resisting  medium,  the  resistance  is  a 
function  of  the  velocity  of  the  body.  If  the  resistance  be 
represented  by  <j>  (v) ,  the  equation  of  motion  becomes 

where  Fis  the  external  force  acting  along  the  right  line. 

It  is   usual  to  assume,   with  Newton,    that   (p  (r)  =  ^v2, 
where  ju  is  a  constant  depending  on  the  density  of  the  me- 
dium and  on  the  area  (8)  of  the  greatest  section  of  the  body 
taken  perpendicular  to  the  direction  of  motion. 
Hence  we  get 


do 

mTt 

=  F-fiv\ 

If  we  suppose 

F  constant, 

and  make 

F 

=  nV\ 

we  get 

dv 

m—  = 

dt 

M(F3-^; 

(1) 

(2) 

If  the  initial  velocity  be  less  than  V,  it  is  obvious  that  the 
velocity  increases  so  long  as  it  is  less  than  V :  this  gives  V 


220  Constrained  Motion. 

as  the  limit  to  which  the  velocity  approaches.     For  this  rea- 
son V  is  called  the  terminal  velocity  of  the  body. 

Also,  since         1  1(1  1      ) 


V2-vz     2V\  V+v      V-vY 
the  preceding  equation  gives 

,      mV,      fV+v\  ._. 

t=Wl0%(v^v)'  ^ 

No  constant  is  added  since  we  suppose  t  reckoned  from 
the  position  of  rest. 

Equation  (3)  shows  that,  while  v  increases  with  t,  yet 
when  v  =  V  we  should  have  t  =  oo  .  Accordingly  the  body 
requires  an  infinite  time  before  arriving  at  its  terminal  ve- 
locity. 

195.  Vertical  Motion. — One  of  the  most  important  cases 
is  that  of  a  body  falling  vertically  in  a  resisting  medium. 
In  this  case  F  =  mg,  and  equation  (3)  becomes 

(4) 

~TT   .     ..  lot 

This  gives 
Hence 

Again,  since 

we  get 

when  x  is  measured  from  the  position  of  rest. 


I' 

V+v 

r-v" 

2yt 
V 

V 

at            gt 

ev-  e  v 

ft         _  st  ~ 

ev+  e'v 
dx 

v  =  Tt> 

Ranh 

gt 

V 

X 

=  7l0<    2 

gt 

z> 

Vertical  Motion.  221 

This  may  be  written  in  the  form 

x  -  —  log  cosh  —  (6) 

Again  we  may  write  /ul  =  AS,  where  A  is  a  constant  de- 
pending on  the  density  of  the  medium. 
Hence  from  (1)  we  get 


£-J 


-£-,  (7) 

AS'  K  } 


where  W  denotes  the  weight  of  the  body. 

This  shows  that,  W  remaining  the  same,  the  value  of  V 
can  be  increased  by  diminishing  the  area  of  the  transverse 
section. 

In  the  case  of  a  homogeneous  sphere  of  radius  r,  we  have 
W=  iirr^p,  where;;  is  the  weight  of  a  unit  of  volume;  also 
S  =  wr*  ;  therefore 

4pr 
SA 


I 


Hence  we  see  that  for  spheres  of  the  same  density  that  of  the 
greater  radius  has  the  greater  terminal  velocity,  and  we  can 
readily  compare  the  vertical  motions  of  different  spheres  in 
the  same  resisting  medium. 

Next,  for  a  body  projected  vertically  upwards  in  a  resist- 
ing medium  the  equation  of  motion  is 


V2     civ 
whence  at  = 


Accordingly,  if  V0  be  the  initial  velocity,  we  find 

t  =  —    tan-1  -77  -  tan  l  — 
g\         V  V 


222  Constrained  Motion. 

From  this  equation  the  velocity  at  any  instant  can  be  de- 
termined. 

Also,  since  v  =  0  at  the  highest  point,  the  time  of  ascent 

to  that  point  is  represented  by  —  tan-1  — . 

V2     vdv 
Again  dx  =  ~Y^TV2' 

Hence,  if  x  be  measured  upwards  from  the  point  of  projec- 
tion, we  have 

_  F%      Fo2  +  V2 
X~2g     g  v*  +  V2  ' 

If  h  be  the  height  of  ascent,  we  get 

^Yg^K—V2-)-  (8) 

If  the  time  t  be  reckoned  from  the  instant  at  which  the  ' 
body  is  at  its  highest  point,  we  have 

*=Ftan^.  (9) 

The  downward  motion  is  given  by  the  former  investigation. 


Examples.  223 


Examples. 

1.  Find  a  vertical  curve  such  that  the  time  of  describing  any  arc,  measured 
from  a  fixed  point,  shall  he  equal  to  that  of  describing  the  chord  of  the  arc. 

Taking  the  origin  at  the  fixed  point,  the  time  down  a  chord  r,  whose  incli- 
nation to  the  vertical  is  0,  as  in  Art.  46,  is 


J: 


•lr 


g  cos0 
Also  the  time  of  descending:  the  arc  is 


V2g 

where  0O  is  the  value  of  0  when  r  =  0. 

Hence,  since  the  times  are  the  same  for  all  chords,  we  get,  by  differentiation, 

dr 

r  sm  0  +  cos  0  — 

(19 


cos  9 


J~£) 


1  dr 
This  gives  -  —  =  cot  26  ; 

°  r  tf0 

hence  we  get  r2  =  a2  sin  20, 

where  a  is  a  constant.     Accordingly  the  curve  is  a  Lemniscate. 

2.  Investigate  the  corresponding  problem  when  the  acting  force  is  propor- 
tional to  the  distance  from  a  fixed  point. 

Let  A  be  the  position  of  the  fixed  point,  0  the  point  of  departure  of  the  par- 
ticle, P  its  position  at  any  instant,  6  —  L  POA,  OA  =  a ;  then  we  find,  without 
difficulty,  that  the  time  h,  of  describing  OP,  when  the  absolute  force  is  taken  as 
unity,  is  given  by 

.     ,  r  —  a  cos  9      ir 

h  =  sm-1  —  +  -  • 

a  cos  9  2 

Also  the  time  of  describing  the  arc  OP  is 

2 


*2  =  J 


Hence,  since  t\  =  h,  we  have 


W-® 


V2ar  cos  0 


tf0. 


i =  d    I  .     ,r-acos9\  9      (dry 


224  Rectilinear  Motion  in  a  Resisting  Medium. 


therefore  (-  r  tan  9 


-J**©' 


from  which  we  get  r2  =  a2  sin  20.  This  represents  a  lemniscate  also,  as  in  the 
previous  question. 

3.  If  the  motion  of  a  conical  pendulum  be  slightly  disturbed,  prove  that  the 
period  of  a  vibration  is  —=■     /-,  and  the  corresponding  apsidal  angle 

a 

it  ,  where  b  is  the  distance  from  the  centre  to  the  plane  of  the  conical 

Vtf3  +  Zb2  r 

pendulum. 

4.  A  particle  is  projected  from  a  given  point  in  a  horizontal  direction  along 
the  surface  of  a  smooth  sphere  ;  find  the  velocity  of  projection  in  order  that  the 
particle  should  rise  to  a  given  height  on  the  surface  before  commencing  to 
descend. 

5.  A  particle  is  constrained  to  move  in  a  smooth  circle,  under  the  action  of  a 
central  force  which  varies  directly  as  the  distance.  If  the  time  of  describing 
any  arc  be  constant,  prove  that  its  chord  envelops  a  circle. 

Townsend,  Eduo.  Times,  1875. 

6.  If  a  particle  describe  a  curve  freely  under  the  combined  action  of  the 
forces  F,  F',  &c,  where  F,  F\  &c,  act  along  r,  r',  &c,  prove  that  the 
equation 

must  be  satisfied  at  every  point  of  the  curve,  where  <p,  <p',  &c,  denote  the  forces 
respectively  co- directional  with  F,  F',  &c,  under  which  singly  the  given  curve 
would  be  described;  and  y,  y',  &c,  are  the  corresponding  semichords  of  the 
circle  of  curvature  at  the  point. 

Curtis,  Messenger  of  Mathematics,  1880. 

Here,  it  is  easily  seen  by  equation  (25),  Art.  162,  that 

v2  =  Fy  +  F'  y  +  &c. 

Also,  by  (13),  Art,  160,          vdv  =  -  Fdr  -  F'dr'  -  &c. 

Hence  2  (Fdy  +  ydF)  +  22Fdr  =  0, 

or  2  {F(dy  +  2dr)  +  ydF }  =  0. 

Hence,  in  particular,  we  have 

<p  (dy  +  2dr)  +  yd<p  =  0,  &c. 

dy  +  2dr      -  d<p      „ 

or  J— =  T,   &c; 

7  <t> 

This  theorem  plainly  contains  as  a  particular  case  that  given  in  Art.  190. 


Examples.  225 

7.  Apply  the  preceding  to  the  case  of  a  conic  described  under  the  action  of 
forces,  F,  F',  directed  to  its  foci. 

TT  A*  »  A4' 

Here  $=-?>     <P  =  ~  >     7  =  7  ; 

r-  r  - 

therefore  —  —  ( Fr2)  dr  +  -~  —  IF'  /2)  dr  =  0 , 

r2  f/r  r*  dr 


or,  since  rfr  +  dr'  =  0, 


-4  £(Ft*)  =  4/r  (^"2). 
r2  ^/-  r 2  dr  ' 


This  is  satisfied  by  the  equations 

^^(i?';'2)=/l(r)+/2(^-")' 
^/^-.(F'r'2)=Mr')+ma-r'), 

where /1  and /a  are  both  arbitrary  functions. 

If  we  assign  the  same  form  (/)  to  /1  and  /2,  we  obtain  as  a  particular 
solution 

F=^-Sr2{f{r)  +f(2a  -  r)}  dr,  &e. 

If  any  particular  form  be  assigned  to/,  a  corresponding  form  of  Fy  as  also 
of  F',  will  result. 

8.  As  an  example  of  the  preceding,  show  that  a  particle  can  be  made  to  de- 
scribe an  ellipse  freely  under  the  action  of  forces, 

A4  ,      A4' 

Ar  +  ~,     Ar  +  — , 

r-  r  - 

directed  to  its  foci. 

The  student  is  referred  to  Professor  Curtis'  Paper  for  additional  applications. 

9.  A  spherical  particle  moves  within  a  smooth  rectilinear  tube,  which  re- 
volves about  one  extremity  with  a  uniform  angular  velocity  in  a  horizontal 
plane  ;  find  the  motion  of  the  particle. 

Let  w  be  the  angular  velocity  of  the  tube,  and  r  the  distance  of  the  particle, 
at  any  time  t,  from  the  fixed  extremity  of  the  tube  ;  then,  since  the  force  acting 
on  the  particle  is  always  perpendicular  to  r,  we  have  (Art.  28), 


d-r  I 

lc--r\ 


d0\2     n  d-r 

—  1  =  0,      or      -rw-  <»'r  =  °- 

dt )         '  df- 


226  Rectilinear  Motion  in  a  Resisting  Medium. 

dr 
Hence  r  =  ce^{  +  c'e-^K     If  r  =  a,  and  —  =  b,  when  £  =  0,  we  get 

2wr  =  (aa>  +  b)  c"*  +  (ctca  -  b)  e-"*. 

10.  Consider  the  same  prohlem  if  the  tuhe  he  supposed  to  revolve  uniformly 
in  a  vertical  plane. 

Here,  if  the  time  he  reckoned  from  the  instant  that  the  tuhe  was  horizontal, 
the  equation  of  motion  is 

— -  -  dP-r  =  -  g  sin  at. 
air 

The  integral  of  this  is 

r  =  Ce^t  +  CVW<  +  -/-z  sin  at, 

and  the  constants  can  he  determined  from  the  initial  conditions. 

11.  Two  spheres  of  the  same  diameter,  hut  of  different  weights,  fall  freely 
in  air  ;  find  the  ratio  of  the  maximum  velocities  they  will  attain,  stating  clearly 
what  assumptions  you  make.  Lond.  Univ.,  1881. 

12.  Explain  what  is  meant  hy  the  terminal  velocity  of  a  hody  in  a  resisting 
medium. 

If  the  resistance  vary  as  the  square  of  the  velocity  and  the  hody  move  in  a 
vertical  line,  prove  that  at  the  time  t,  reckoned  from  the  instant  at  which  the 
hody  is  at  its  highest  position,  its  depth  x  helow  this  position  is  given  hy 


rhen  ascending,  and  by 


co*.  gt 

x  =  —  log  sec  — , 


x  —  —  log  cosh  — , 


when  descending ;  u>  denoting  the  terminal  velocity  in  the  medium. 

Lond.  Univ.,  1883. 

13.  If  a  hody  be  projected  vertically  upwards  in  a  resisting  medium  with  its 
terminal  velocity  for  the  medium,  determine  the  height  of  its  ascent,  and  the 
time  of  reaching  the  highest  point. 

Prove  that,  if  an  engine  can  pull  a  train  of  W  tons  at  a  velocity  V  on  the 
level,  against  resistances  varying  as  the  square  of  the  velocity,  the  engine  exert- 
ing a  constant  pull  of  P  tons :  then  up  an  incline  o  to  the  horizon  the  maximum 
velocity  will  fall  to  FV(1  -  W  sin  a  I P),  and  that  down  the  incline  without 
steam  the  terminal  velocity  is  FV(  JFsina  /  P). 

Prove  that,  if  on  a  long  railway  journey,  performed  with  average  velocity  V, 
the  actual  velocity  v  varies  from  its  mean  value  by  a  periodic  function  of  the 
time,  say  v  =  V+  Usinnt,  the  average  horse-power  and  consumption  of  fuel  is 
to  that  required  to  take  the  train  with  uniform  velocity  V  as 

1  +  |  Z72  /  V°-  :  1. 

Lond.  Univ.,  1887. 


(     227    ) 


CHAPTER  IX. 


THE  GENERAL  DYNAMICAL  PRINCIPLES. 

196.  D'Alembert's  Principle. — If  a  system  of  mate- 
rial points  connected  together  in  any  way,  and  subject  to  any 
constraints,  be  in  motion  under  the  influence  of  any  forces, 
each  point  of  the  system  has  at  any  instant  a  certain  accele- 
ration. If  now  to  each  point  an  acceleration  were  applied 
equal  and  opposite  to  its  actual  acceleration,  the  velocities  of 
all  the  points  of  the  system  would  become  constant — in  other 
words,  each  point  would  move  as  if  free  and  unacted  on  by 
any  force  whatever ;  that  is,  the  applied  accelerations,  the 
external  forces,  and  the  constraints  and  mutual  or  internal 
forces  of  the  system,  would  equilibrate  each  other. 

Stated  in  algebraical  language,  the  principle  which  is 
given  above  may  be  enunciated  as  follows : — If  the  coordi- 
nates of  any  particle  m  of  a  material  system  be  a?,  //,  z,  and 
the  external  forces  there  applied  X,  Y,  Z;  the  system  of 
forces, 

d2xx  d~y,  d2z, 

Xl"Wl^'     Y'~nh-dF>    Zl-mi~df> 

d2x2  d2y2  d2z% 

x*-nhHF>     Y*-,)hliF>     ^->^>&c-> 

acting  at  the  points  %it/iZu  x2y2z:,  &c,  will  be  in  equili- 
brium, in  virtue  of  the  constraints  and  mutual  reactions  of 
the  system. 

d2x  d2t/         (Pz 


The  force  whose  components  are  -  m  — ,  -  m  , 
1  dt?9  dt 


2> 


m 

d 


is  called  the  force  of  inertia  of  the  mass  m,  and  D'Alembert's 
Principle  (as  stated  in  Article  71)  simply  expresses  that — 

The  applied  forces  and  the  forces  of  inertia  in  any  system  are 
in  equilibrium. 

q2 


228  The  General  Dynamical  Principles. 

In  applying  D'Alembert's  Principle,  we  may,  as  in  Statics, 
consider  the  constraints  of  the  system  either  as  geometrical 
conditions,  or  else  substitute  for  them  unknown  forces.  In 
the  algebraical  statement  just  given,  the  former  plan  has 
been  adopted ;  but  if  we  choose  to  adopt  the  latter,  we  have 
merely  to  make  X,  Y,  Z,  &c,  include  not  only  the  applied 
forces,  but  also  the  stresses  arising  from  the  constraints. 

If  the  Statical  Principle  of  Yirtual  Yelocities  be  employed, 
we  have  for  D'Alembert's  Principle  the  concise  mode  of 
expression  given  by  Lagrange  in  his  Mecanique  Analytique, 
viz. : — 

s  j(x-.g)fc+(p:-«S)*r+^-S)fcJ-o.  (1) 

This  equation  may  also  be  written 

Sw  [xlx  +  j/By  +  zBz)  =  S  {XSx  +  YSy  +  Z$z),      (2) 

a  form  which  is  often  more  convenient  than  (1). 

If  the  forces  X,  Y,  Z,  &c,  constitute  a  conservative 
system,  Art.  124,  we  may  write 

2(X&-+  YSy  +  ZSz)  =SY, 

and  (2)  becomes  in  this  case 

Sw2  (£&  +  ydy  +  zh)  =  SY.  (3) 

197.  D'Alembert's    Principle    for    Impulses. — As 

has  been  stated  already  in  Article  66,  an  Impulsive  or  In- 
stantaneous Force  is  a  force  which  produces  a  finite  change 
of  velocity  in  a  time  so  short  that  in  it  no  sensible  change 
of  velocity  is  produced  by  the  action  of  the  forces  which  are 
not  impulsive.  If  the  constraints  and  connections  of  a  system 
be  regarded  as  giving  rise  to  forces,  these  forces  may  be  im- 
pulsive or  not,  according  to  the  nature  of  the  constraint.  For 
example,  a  blow  given  to  a  body  which  is  resting  on  an  im- 
movable surface  produces  an  impulsive  reaction,  provided  the 
blow  is  not  tangential  to  the  surface  ;  but  a  sudden  jerk  to  a 
body  attached  to  the  end  of  an  extensible  elastic  string  pro- 
duces no  impulsive  reaction.     It  is  important  to  observe  that 


D'AIembert's  Principle  for  Impulses.  229 

each  point  of  the  system  may  be  regarded  as  occupying  the 
same  position  in  space  at  the  end  as  at  the  beginning  of  the 
time  during  which  the  impulsive  forces  have  acted.  In  other 
words,  the  velocities  of  the  various  points  may  change  by  a 
finite  amount,  but  the  positions  can  only  change  by  an  infi- 
nitely small  amount  during  the  time  under  consideration. 

If  u,  v',  w  be  the  components  of  the  velocity  of  any 
point,  whose  coordinates  are  a?,  y,  s,  before  the  action  of  the 
impulsive  forces;  and  u,  v,  w  the  corresponding  velocities 
after  their  action  ;  and  X,  F,  Z  be  the  components  of  the  im- 
pulse which  has  acted  at  this  point,  D'AIembert's  Principle 
as  applied  to  impulsive  forces  may  be  expressed  in  the  form — 

Sm ( [u  -  u') Sx  +  [v  -  v')  hj  +  (w  -  w)  Ss)  =  2  [XBx  +  Ydy  +  ZSz). 

(4) 

The  truth  of  the  Principle  in  the  present  case  can  be 
established  by  reasoning  similar  to  that  employed  in  the 
preceding  Article. 

It  may  also  be  derived  from  the  Principle  applied  to 
continuous  forces,  by  considering  the  impulsive  forces  as 
continuous  forces  of  great  magnitude  acting  for  a  very  short 
time.     In  fact,  if  we  multiply  >the  equation 

-S)a.+.(r-^^+(*-«S)*|-° 

by  dt9  and  integrate  between  the  limits  t  and  f;  if  the  interval 
t-t'he  sufficiently  short,  the  system  has  not  sensibly  altered 
its  position,  and  therefore  Sa>,  &c,  are  the  same  at  the  end  of 
the  time  as  at  the  beginning,  and  we  have 


V 


(dx  _  fdx\\] 


_jt 


Xdt-m\Tt-{jt  njfc+M-o- 


Now,  if  X  be  the  component  of  a  continuous  force,    ^  Xdt 

is  insensible ;  and  if  Xbe  the  component  of  an  impulsive  force, 

it 
Xdt  is  the  component  of  the  impulse  along  the  axis  of  x, 


230  The  General  Dynamical  Principles. 

which  may  be  denoted  by  X ;  hence,  as 

dx  fdxY       ,    0 

we  immediately  obtain  equation  (4). 

198.  Initial  Motion. — If  a  system  start  from  rest 
under  the  action  of  given  impulses,  equation  (4),  Art.  197, 
becomes 

2w  {uSx  +  v$y  +  wdz)  =  2  (XSx  +  Tdy  +  Zdz),       (5) 

where  u,  v,  w  are  the  components  of  the  initial  velocity  of  the 
point  xyz.  Now  as  Bx,  Sy,  $z  are  any  arbitrary  displacements 
of  this  point,  consistent  with  the  conditions  of  the  system,  we 
may,  if  the  equations  of  condition  do  not  involve  the  time 
explicitly,  substitute  for  8x,  Sy,  §z  the  actual  displacements  of 
the  point  (see  Art.  200).  Hence,  as  actual  displacements 
when  divided  by  the  element  of  time  become  velocities,  we 
may  substitute  for  Bx,  By,  Bz  the  components  u' ,  v',  w',  of  the 
velocity  of  xyz  in  any  actual  motion  of  the  system.  Thus  we 
obtain 

2m  {uu  +  vv  +  ww)  =  S (Xu'  +  Yv'  +  Zw').         (6) 

Examples. 

1.  If  the  same  system  be  set  in  motion  successively  by  two  different  im- 
pulses applied  at  the  same  point,  each  impulse  is  proportional  to  the  velocity  in 
the  direction  of  the  other  which  it  imparts  to  its  point  of  application. 

Let  these  velocities  be  q  and  jt/,  and  let  X,  Y,  Z;  X',  Y',  Z'  be  the  compo- 
nents of  the  impulses  P  arid  Q,  and  u,  v,  w  ;  u',  v',  w'  the  components  of  the 
initial  velocities  of  the  point  of  application,  then, 

Xu'  +  Yv'  +  Zw'  =  ~Zm{uu  +  vv'  +  ww)  =  X'u  +  Y'v  +  Z'w  ; 

but  Fp  =  Xu'  +  Yv'  +  Zw',     and     Qq  =  X'u  +  Y'v  +  Z'w, 

whence  P  :  Q::g  :  p'. 

2.  In  any  system  at  rest,  if  we  suppose  an  impulse  P  applied  at  a  point  A, 
and  an  impulse  P'  applied  at  a  point  B ;  prove  that 

P:  P'  =  v  :v\ 

where  v  is  the  component,  in  the  direction  of  P',  of  the  velocity  of  the  point  B 
due  to  the  impulse  P ;  and  v'  is  the  similar  component  of  velocity  of  the 
point  A. 


Energy  of  Initial  Motion.  231 

199.  Energy  of  Initial  Motion.— If  T  be  the  initial 
kinetic  energy  of  a  system  set  in  motion  by  given  impulses, 
by  substituting  u,  v,  w  for  &r,  hj,  $z  (in  5)  we  obtain 

2T  =  Sw  O2  +  v2  +  w2)  =  2  (Xu  +  Yd  +  Zw).         (7) 

BertramVs  Theorem* — If  a  system  start  from  rest  under 
the  action  of  given  impulses,  every  additional  constraint 
diminishes  the  initial  kinetic  energy. 

Let  ii ',  v\  w  be  the  initial  velocities  of  the  point  xyz  under 
the  action  of  the  given  impulses  when  the  additional  con- 
straints are  imposed ;  and  u,  v,  w  the  initial  velocities  when 
the  system  is  free  from  these  constraints,  then,  u'dt,  v'dt,  w'dt 
are  possible  displacements  in  the  unconstrained  as  well  as  in 
the  constrained  system.  Hence,  substituting  u',  v,  w  for 
&r,  $y,  cz  in  equation  (5)  we  obtain 

2w  (uu  +  vv  +  ww')  =  2  (Xu'  +  Yd'  +  Zw'). 

But,  by  (7),     2w  (u'°~  +  v'2  +  w'2)  =  2  [Xu*  +  Yv'  +  Zw)  ; 
thus  we  have 

Sm  { (u  -  it)"  +  (v  -  v'y  ■¥  (to  -  w'Y\  =  "2m  (u2  +  v2  +  iv2) 

-  22w  (uu'  +  vv'  +  ww')  +  "2m  (it2  +  v'2  +  w2) 

=  2T-  4T'  +  2T'  =  2T-  2T'.  (8) 

Hence,  we  see  that  the  energy  of  the  unconstrained 
exceeds  that  of  the  constrained  motion  by  the  energy  of  the 
motion  which  must  be  combined  with  either  to  produce  the 
other. 

Thomson's  Theorem. f — If  impulses  are  applied  only  at 
points  where  the  velocities  are  prescribed,  additional  con- 
straints increase  the  initial  kinetic  energy. 

Here,  when  additional  constraints  are  imposed,  the  im- 
pulses are  supposed  to  be  altered  in  such  a  manner  as  still 
to  produce  the  prescribed  velocities  in  the  assigned  points  ; 
then,  u',  v\  w  being,  as  before,  the  velocities  belonging  to  the 
constrained  motion,  we  have,  since  in  the  present  case  the 

*  Liouville,  tome  septieme  (1842),  p.  165. 

t  Proceedings  of  Royal  Society  of  Edinburgh,  April,  1863. 


232  The  General  Dynamical  Principles. 

velocity  of  every  point  at  which  an  impulse  acts  is  sup- 
posed to  remain  unaltered, 

S  {Xu'  +  IV  +  Ziv)  =  S  [Xu  +Yv+  Zw)  =  2T. 

Hence  by  (6)  we  obtain 

Sm  { [ii  -  uf  +  {vf  -  vf  +  (to'  -  ivf }  =2T'-2T,     (9) 

and  therefore  T  exceeds  T  by  the  energy  of  the  additional 
motion. 

Examples. 

1.  A  system  is  set  in  motion  by  an  impulse  which  is  measured  by  the 
momentum  of  a  mass  of  60  lbs.  moving  with  a  velocity  of  24  feet  per  second. 
The  impulse  imparts  to  its  point  of  application  a  velocity  of  8  feet  per  second  in 
a  direction  inclined  to  that  of  the  impulse  at  an  angle  of  60°.  Find  in  foot 
pounds  the  initial  kinetic  energy  of  the  system.  Ans.  90. 

2.  If  the  initial  velocities  of  certain  points  of  a  system  be  given,  prove  that 
its  initial  kinetic  energy  is  least  when  tbe  system  is  set  in  motion  by  impulses 
passing  through  these  points. 

200.  Equation  of  Vis  Viva.— A  first  integral  of  the 
equations  of  motion  can  very  frequently  be  obtained  directly 
from  D'Alembert's  Principle,  as  follows  : — 

where  &r,  hj,  Bz,  &c.  are  arbitrary  displacements  consistent 
with  the  conditions  of  the  system.  If  the  equations  of  con- 
dition do  not  contain  the  time  explicitly,  dx  (the  actual 
movement  of  the  point  along  the  axis  of  x  during  an  infinitely 
short  time)  is  always  a  value  which  may  be  legitimately 
assigned  to  Sa? ;  for  the  fact  that  it  is  an  actual  displacement 
shows  that  it  is  consistent  with  the  equations  of  condition, 
and  therefore  possible,  provided  these  equations  do  not  alter 
with  the  time  ;  that  is,  do  not  contain  the  time  explicitly.  If 
they  contain  the  time  explicitly,  dx  is  not  in  general  a 
possible  value  of  &e.  In  fact,  IT  =  0  being  an  equation  of 
condition,  if  U  is  a  function  of  the  coordinates  simply,  &r 
and  dx  must  satisfy  the  same  equation,  viz., 

dx  dy 


Equation  of  Vis  Viva.  233 

If,  however,  TJ  contains  t  explicitly,  $x  has  to  satisfy  the 
equation 

4?&e  +  &o.  =  0, 
ax 

where  t  is  treated  as  constant;  but  dx  has  to  satisfy  the 
equation 

-7-  dx  +  &c.  +  — -  at  =  0. 
dx  at 

This  is  so,  because  dx  is  the  interval  between  two  successive 
positions  of  a  point,  at  consecutive  instants  of  time ;  whereas 
he  is  the  interval  between  two  simultaneous  infinitely  near 
possible  positions  of  the  point. 

In  the  great  majority  of  problems  dx  is  a  possible  value 
of  Sx ;  and  the  same  holds  for  the  other  displacements. 
Assuming  then  that  the  transformation  is  legitimate,  let  us 
assign  to  Sx,  8y,  &c.  the  values  dx,  dp,  &c. ;  D'Alembert's 
equation  becomes  then 

2ra  (~  dx  +  ^<ty+f??dk)  =  2  {Xdx  +  Tdy  +  Zdz). 
Integrating,  we  have   . 

where  c  is  an  arbitrary  constant. 

This  equation  is  called  the  equation  of  vis  viva. 

If  we  denote  the  vis  viva  at  any  particular  time  t'  by 
2wp'2,  the  equation  above  may  be  written — 

2»w2  -  2m/2  =  22 / (Xdx  +  Ydy  +  Zdz),         (11) 

where  the  right-hand  side  is  twice  the  work  done  by  the 
forces  in  passing  from  the  position  occupied  at  the  time  ^to 
the  position  occupied  at  the  time  t.  For  a  conservative 
system  (11)  becomes 

T  -  T  -  Y  -  Y.  (12) 

If  there  be  no  forces  acting  on  the  system  its  vis  viva 
remains  constant. 


234  The  General  Dynamical  Principles. 

The  equation  of  vis  viva  has  been  already  obtained  for  a 
rigid  body  in  a  different  manner  in  Article  132. 

The  equation  of  vis  viva  is  one  of  the  most  important  in 
Dynamics,  and  is  to  a  great  extent  the  foundation  of  the 
Theory  of  Energy.  It  will  be  more  fully  considered  in  a 
future  chapter. 

201.  Of  the  Forces  which  enter  the  Equation  of 
Wis  "Viva. — From  the  mode  in  which  the  equation  of  vis  viva 
has  been  deduced  from  D'Alembert's  Principle,  it  is  plain 
that  in  the  case  of  a  rigid  body  the  right-hand  side  contains 
only  the  applied  forces,  and  that  reactions  by  which  geome- 
trical conditions  may  be  replaced  do  not  enter  therein.  The 
reactions  of  fixed  points,  fixed  surfaces,  &o.,  are  thus  ex- 
cluded :  and  farther,  if  during  the  motion  the  direction  of  a 
force  be  at  each  instant  at  right  angles  to  the  line  in  which 
its  point  of  application  is  moving,  such  a  force  does  not  enter 
the  equation  of  vis  viva  (Art.  122). 

When  two  surfaces  roll  on  one  another  without  slipping, 
the  relative  tangential  displacement  of  the  two  points  in  con- 
tact is  zero.  Now,  if  F  be  the  tangential  force  of  friction 
developed  between  them,  the  element  of  work  done  by  F  on 
one  body  is  Fdfx,  and  that  done  on  the  other  is  -  Fdf2,  dfy  and 
df2  being  the  projections  on  the  direction  of  i^of  the  small 
motions  of  the  two  points  in  contact.  Hence  the  whole  work 
done  by  the  tangential  friction  is  F(dfl  -  clf2)  ;  but  df\  -  df2 
is  the  relative  tangential  displacement  of  the  points  of  the 
surfaces  which  are  in  contact.  Hence  the  whole  work  done 
by  the  tangential  force  of  friction  in  pure  rolling  is  zero,  or 
this  force  in  the  case  supposed  has  no  effect  on  the  equation 
of  vis  viva.  If  one  of  the  surfaces  be  fixed,  a  similar  result 
obviously  holds  good. 

"When  two  surfaces  are  in  permanent  contact,  the  normal 
reaction  between  them  never  enters  the  equation  of  vis  viva. 
For,  if  the  motion  be  pure  slipping,  the  relative  velocity  of 
the  points  of  application  of  the  mutual  reaction  is  altogether 
tangential.  If  the  motion  be  either  rolling  and  slipping,  or 
pure  rolling,  the  relative  normal  velocity  of  the  points  in 
contact  is  still  zero,  or  at  least  infinitely  small,  and  the  rela- 
tive normal  displacement  is  an  infinitely  small  quantity  of 
the  second  order  ;  that  is,  d)\  -  dr2  =  0,  where  d)\  and  dr2  are 


Effect  of  Impulses  on  Vis  Viva.  235 

the  projections  on  the  common  normal  of  the  small  motions 
of  the  points  in  contact.  Hence  R  (d)\  -  dr2),  which  is  the 
whole  work  done  by  the  mutual  normal  reaction,  is  equal  to 
zero. 

The  work  done  in  the  element  of  time  by  a  mutual  force 
between  two  bodies  is  always  of  the  form  R  [drx  -  dr2),  where 

— -  and  —jj  are  the  velocities,  in  the  direction  of  the  joining 

at  Clt 

line,  of  the  points  between  which  the  force  R  acts.  If  r  be 
the  distance  between  these  points,  the  work  done  by  the 
mutual  action  is  therefore  R  dr. 

If  R  tends  to  increase  the  relative  velocity  in  its  own 
direction  which  already  exists,  Rdr  is  positive.  If  on  the 
other  hand  R  tends  to  diminish  this  velocity,  R  dr  is  negative. 
This  readily  appears  from  the  following  considerations  : — 

If  the  mutual  action  tends  to  increase  the  velocity  — ,  it 
tends  to  diminish  — 2,  and  therefore  the  element  of  work  done 

0.0 

by  it  is  R  (di\  -  dr2) ;  but  this  is  positive  if  —  >  — ,   and 

clt       c(t 

negative  if  ~  >  — ~.      In  the  first  case  the  mutual  action 
5  dt        clt 

tends  to  increase  the  relative  velocity  in  its  own  direction, 
and  in  the  second  case  to  diminish  this  velocity.     Also,  if 

the  mutual  action  tends  to  diminish  — -1   similar    reasoning 

applies. 

202.  Effect  of  Impulses  on  lis  Viva, — The  change 
of  vis  viva  resulting  from  impulses  may  be  investigated  by 
means  of  equation  (4),  Art.  197. 

In  general  for  any  displacement  c\  whose  components 
are  Ex,  $y,  dz,  we  have  X  Sx  +  Yhj  +  Z%z  =  R  or,  where  R  is 
the  impulse  whose  components  are  X,  Y,  Z,  and  oY  is  the 
projection  of  cs  on  the  direction  of  R.  Hence,  if  the  direc- 
tions of  R  and  &s  are  at  right  angles  to  each  other, 

XSx+  Y§y+Z$z=  0. 


236  The  General  Dynamical  Principles. 

Again,  if  two  equal  and  opposite  impulses  occur  at  points 
whose  coordinates  are  Xi,  yXi  zx ;  x2,  y2,  z2,  the  corresponding 
terms  in 

2(X&r+F8y  +  Z&) 

are  X  {Sx,  -  $x2)  +  F(S^  -  fy2)  +  Z  (dz,  -  &,), 

or  R  (h\  -  Sr2),  from  which  we  conclude,  that  if  the  relative 
displacement  of  two  points  be  perpendicular  to  the  direction 
of  the  mutual  impulsive  reaction  at  those  points,  the  corre- 
sponding terms  in  2  (XSx  +  TBy  +  ZSz)  vanish. 

We  can  now  prove  the  following  theorems :  — 

If  a  system  be  acted  on  by  external  impulses,  the  vis  viva 
is  diminished  by  the  vis  viva  of  the  additional  motion  when 
the  impulse  at  each  point  is  perpendicular  to  the  subsequent 
velocity  of  that  point,  but  increased  by  the  same  amount 
when  the  impulse  is  perpendicular  to  the  antecedent  velocity. 

Similar  results  hold  good  for  internal  impulsive  reactions 
when  each  mutual  impulse  is  perpendicular  to  the  relative 
velocity  of  the  points  between  which  it  acts. 

For  the  same  notation  being  adopted  as  in  Art.  197 — 
1°  when  each  impulse  is  perpendicular  to  the  subsequent  ve- 
locity of  the  point  at  which  it  acts,  we  have 

2{Xu+  Yv  +  Zic)  =  0; 

and  2°  when  it  is  perpendicular  to  the  antecedent  velocity, 

S(Ij/+  Yv'+  Zw=  0. 

In  the  first  case,  substituting  u,  v9  w  for  Ex,  By,  Bz  in  equa- 
tion (4),  we  get 

2  m  [  (u  -  u')  u  +  (v  -  v)  v  +  (w  -  vf)  w }  =  0 ; 

from  which  we  obtain 

2  m  ( [u  -  uj  +  [v  -  vj  +  (iv  -  to')2} 

=  2 m  (u2  +  v'2  +  w2)  -  2 m (u2  +  v2  +  to2).  (13) 

In  the  second  case,  substituting  u',  v\  w  for  $x,  By,  Sz  in 
(4),  we  obtain  in  like  manner 

2  7ft  { (u  -  uf  +  (v  -  v)2  +  (w  -  tof) 

=  2 m (n2  +  v2  +  w2)  -  2 m [u2  +  v2  +  w2).  (14) 


Effect  of  Impulses  on  Vis  Viva.  237 

Bertrand's  Theorem  (Art.  199)  is  obviously  included  in 
the  first  case  of  the  above  theorems. 

The  impulses  resulting  from  the  impact  of  inelastic  bodies 
against  fixed  obstacles,  or  against  one  another,  as  well  as 
those  produced  by  sudden  pulls  on  inextensible  strings,  come 
under  the  first  case  considered  above.  To  the  second  case,  on 
the  other  hand,  belong  impulses  due  to  explosions,  or  to  the 
process  of  restitution  which  takes  place  in  the  second  period 
of  the  impact  of  elastic  bodies. 

As  has  been  already  stated  (Art.  78),  in  the  impact  of 
such  bodies  there  are  two  periods.  In  the  first,  the  mutual 
action  reduces  the  relative  normal  velocity  of  the  colliding  points 
to  zero.  In  the  second,  a  force  of  restitution  is  developed, 
which  acts  at  each  point  in  the  same  direction  as  the  original 
force,  and  produces  an  impulse  which  bears  a  constant  ratio 
to  that  belonging  to  the  first  period.  This  constant  ratio  is 
called  the  coefficient  of  restitution. 

As  the  special  equations  which  determine  the  changes  of 
velocity  in  terms  of  the  corresponding  impulses  are  obtained 
by  equating  to  zero  the  coefficients  of  the  independent  varia- 
tions in  equation  (4),  Art.  197,  we  see  that  these  equations 
are  always  linear.  Moreover,  in  the  impact  of  elastic  bodies 
the  geometrical  conditions  are  the  same  in  the  periods  of 
compression  and  of  restitution ;  but  each  impulse  in  the  latter 
period  is  equal  to  the  corresponding  impulse  in  the  former 
multiplied  by  the  coefficient  of  restitution.  Hence  we  con- 
clude that  this  holds  good  likewise  for  the  corresponding 
changes  of  velocity  in  the  two  periods. 


Examples. 

1.  If  a  system  be  acted  on  by  any  set  of  impulses,  prove  that  the  increase  of 
its  vis  viva  maybe  expressed  in  the  form  2{  X(u  +  u')  +  Y{v  +-  v)  +  Z(w  +  w')}t 
where  X,  Y,  Z  are  the  components  of  the  impulse  acting  at  any  point  of  the 
system;  and  «,  v,  w,  u',  v',  w'  the  components  of  the  velocity  of  this  point 
after  and  before  the  impulsion,  respectively. 

2.  If  a  system  be  acted  on  by  a  set  of  impulses  which  reduce  to  zero  the 
velocities,  in  the  direction  of  the  impulses,  of  the  points  at  which  they  act,  find 
the  change  of  vis  viva  in  terms  of  the  impulses  and  the  antecedent  velocities  of 
the  points  at  which  they  act.  Ans.  ^{Xu'  +  IV  +  Zw). 


238  The  General  Dynamical  Principles. 

3.  The  ends  of  a  string  passing  over  a  smooth  pulley  are  attached  to  two 
masses,  of  which  one  rests  on  a  horizontal  plane,  and  the  other  is  dropped 
through  a  height  h ,  the  masses  of  the  string  and  pulley  heing  neglected, 
determine  the  loss  of  kinetic  energy  caused  hy  the  impulsive  tension  of  the 
string. 

If  in  and  in'  he  the  masses,  v\  and  v>  the  velocities  of  the  dropped  mass  m 
hefore  and  after  the  chuck,  and  7  the  loss  of  kinetic  energy, 

mm 

2,7  =  m iv\  —  ro)2  +  m  vz2.     Hence  7  =  - ah. 

v  '  m  + m J 

4.  If  any  system  of  smooth  imperfectly  elastic  hodies  having  a  common 
coefficient  of  restitution  collide,  show  that  the  loss  of  vis  viva  is 

1  —  e 

2m  {(u-u')2  +  (v  -v')2  +  (to  -  tv')2}, 

1  +  G 

where  e  is  the  coefficient  of  restitution,  m  the  mass  of  any  particle,  and  u\  v', 
to',  u,  v,  w  the  components  of  its  velocity  hefore  and  after  the  shock. 

Let  U,  V,  Whe  the  components  of  the  velocity  of  in  at  the  end  of  the  first 
period  of  impact;  then  by  equations  (13)  and  (14),  Art.  202,  if  7  be  the  total 
loss  of  kinetic  energy, 

27=2m{(U-u')*  +  {r-v')i+{W-w')*}-2m{(u-U)*+(v-jrf+(w-W)*}; 

but        u  —  U=e(U—  n'),  &c,      and  therefore,        u  —  u'  =  (1  +e)(U—  u'),  &c. 

Hence,  27  =  (1  -  e2)  Sm { ( U- u'f  +  ( V-  v')2+  { W-  w')2} 

1  — '  e2 
=      — y2  Smj{  (m  -ti')2  +  {v  -v')2  +  [w-w')2} . 

The  theorem  contained  in  this  Example  is  due  to  Carnot. 

5.  Two  weights  are  connected  by  a  fine  inextensible  string  passing  over  a 
smooth  pulley.  The  lesser  hangs  vertically,  and  the  other  descends  a  smooth 
inclined  plane,  starting  without  initial  velocity  from  a  point  vertically  under 
the  pulley  ;  determine  how  far  it  will  descend  ;  and  state  the  limit  of  the  ratio 
of  the  weights  within  which  finite  descent  is  possible. 

If  z  be  the  height  which  the  lesser  weight  W  ascends,  and  s  the  distance 
along  the  inclined  plane  traversed  by  the  greater  weight  W,  we  have,  by  the 
equation  of  vis  viva,  when  the  system  comes  to  rest,  TVs  sin  i  -  W'z  =  0,  and 

W 
therefore  z  =  \s  sin  i,  if  —  =  A.      Also,  if  h  be  the  vertical  distance  of  the 

pulley  from  the  inclined  plane,  we  have  geometrically,  since  the  string  is  inex- 
tensible, 

to       o      ^t     •     •      TT  2(A— l)sine, 

(h  -t  z)~  =  //-  -f  s-  -f  2hs  sm  x.     Hence  s  =  -^ .        h. 

1  —  Xz  sin-i 

In  order  that  finite  descent  should  be  possible,  W  >Wrsin  i. 

6.  Two  equal  spheres,  A  and  B,  starting  simultaneously  from  rest,  descend 
down  two  equally  inclined  planes ;  the  one  plane  quite  smooth,  the  other 
perfectly  rough  ;  find  the  ratio  of  the  velocities  of  the  centres  of  the  spheres 
at  the  end  of  any  time. 

Let  v\  be  the  velocity  of  the  centre  of  A,  v%  that  of  the  centre  of  B,  and  w 
the  angular  velocity  of  B  at  the  end  of  any  time  t,  then  v%  —  aw  (see  Ex.  1, 


Examples.  239 


7 
Art.  134),  also,  mvi2=2ffmzi,  and-  ma2  »2  =  2gmz2,  where  z\  and  z2  are  the 

5 

distances  through  which  the  centres  of  A  and  B  have  descended.  Now,  if  S] 
and  s2  he  the  distances  which  the  centres  have  moved  parallel  to  the  inclined 
plane  at  any  time, 

ds\  da*  .     .  .     . 

Vi  = — ,      t'2= — ,     and  :i  =  *i  sin  ?,     r2  =  S2smi. 
dt  dt 

Hence,  substituting  and  differentiating,  we  have 

dn'Si          .    .     7  d'2s2  •     •       tt  5 

- —  =  <?  sin  i, =  q  sin  ^.      Hence  02  =  =«i- 

7.  A  thin  uniform  rod,  AB,  slides  down  between  a  vertical  and  a  horizontal 
rod,  to  which  it  is  attached  by  small  smooth  rings  ;  find  the  angular  velocity  of 
AB  in  any  position. 

Take  the  horizontal  and  vertical  rods  as  axes  of  x  and  y,  their  intersection 
being  0,  and  let  0  be  the  angle  which  AB  makes  with  the  vertical  rod  at  any 
time ;  then  M,  the  middle  point  of  AB,  describes  a  circle  round  0  with  an 
angular  velocity  0,  which  is  likewise  the  angular  velocity  of  the  rod  round  M. 

a2 
Hence  (Art.  134),  ma2  02-f  m  "jj"  02  =  2gma  (cos  a  -  cos0),  where  a  is  half  the 

length  of  the  rod,  m  its  mass,  and  a  the  initial  value  of  0 ;  then 

3ff 
0-  =  —  (cos  a  —  cos0). 

2«  , 

8.  A  narrow  smooth  semicircular  tube,  whose  radius  is  a,  is  fixed  in  a  vertical 
plane,  the  vertex  of  the  semicircle  being  its  highest  point ;  a  heavy  flexible  string 
passing  through  the  tube  hangs  at  rest ;  if  the  string  be  cut  at  one  end  of  the 
tube,  find  the  velocity  which  the  longer  portion  will  have  attained  when  leaving 
the  tube. 

Let  I  be  the  length  of  each  of  the  portions  of  string  which  hang  below  the 
ends  of  the  tube  in  the  position  of  equilibrium ;  then,  since  the  distance  of  the 

.2a  .         ,. 

centre  of  inertia  of  a  semicircular  arc  from  the  centre  is  — ,  a  being  the  radius, 

7T 

we  have,  if  v  be  the  velocity  with  which  the  string  leaves  the  tube, 

(  (2a      Tra\  ) 

(Z  +  7m)*;2=2/7    lira  +  Tra  (  —  +  —  1    , 

whence 

2,/+(4  +  ,> 

l  +  ira  J 

9  A  uniform  bar,  of  length  2a,  is  suspended  from  a  fixed,  parallel,  and  equal 
horizontal  bar,  by  strings  of  equal  length  joining  the  adjacent  extremities  of  the 
bars  An  angular  velocity  a>  is  imparted  to  the  snspended  bar  round  a  vertical 
axis  through  its  centre  of  inertia.  Determine  the  vertical  height  through  which 
its  centre  of  inertia  will  rise. 

As  each  extremity  of  the  bar  moves  on  the  surface  of  a  sphere  to  wnicn  tne 
attached  string  is  radius,  the  tensions  of  the  strings  do  not  appear  in  the  equation 

a2  co2 
of  vis  viva  ;  hence  h  =  —  - . 


240 


The  General  Dynamical  Principles. 


203.  The  General  Equations  of  Motion  of  a 
Rigid  Body  apply  to  every  System. — If  the  forces 
acting  on  any  system  be  in  equilibrium,  the  equilibrium  is 
not  disturbed  by  rendering  the  mutual  distances  of  the  points 
of  the  system  invariable — in  other  words,  by  making  it  rigid. 
Hence  the  equations  of  motion  of  a  rigid  body  are,  in  their 
most  general  form,  applicable  to  any  system  whatever.  The 
special  reductions  which  may  be  applied  to  the  forces  of 
inertia  in  the  case  of  a  rigid  body  cannot,  however,  be  em- 
ployed in  other  cases. 

204.  Equations  of  Motion  of  a  Rigid  Body. — 
By  means  of  D'Alembert's  Principle  we  can  at  once  write 
down  the  equations  of  motion  of  a  rigid  body.  We  have,  in 
fact,  merely  to  write  down  the  six  equations  of  equilibrium, 
taking  into  account,  not  only  the  applied  forces,  but  also  the 
forces  of  inertia  as  denned  in  Art.  196. 

Hence  the  six  equations  of  motion  are — 


(15) 


»£-2X,    *.fjf-ar.    a-9-s* 

~(*£-S)-*<i*-*>-^ 

^(.S-S)-^--*)-* 

>, 

^iUd^-yd^\=^{xY-yX)=N 

(16) 


where  Z,  M,  N  are  the  moments  of  the  applied  forces  round 
the  axes. 

For  impulses,  the  corresponding  equations  are — 

^m [y{w - vf)  -z(v-v')}  =  ^(yZ-zY)=L    \ 
^m{z{u-v!)-x{w-w')}  =  2{zX-xZ)  =  Jf  >  •    (18) 
2m  [x  (v  -v)-y  (u  -  u) }  =  2  [x  Y-  yX)  =  N  J 
These   equations  hold   good  for   any   system    which    is 


Constraints  and  Partial  Freedom. 


241 


altogether  free,  i,  e.  unacted  on  by  any  constraints,  and  not 
subject  to  geometrical  conditions  external  to  itself. 

In  the  case  of  a  system  subject  to  external  constraints, 
the  constraints  must,  in  general,  be  replaced  by  the  stresses 
to  which  they  give  rise. 

Equations  (15)  and  (17)  may  be  put  into  a  simpler  shape 
as  follows : — 

205.  Motion  of  the  Centre  of  Inertia  of  a  Free 
System. — Let  x,  y,  z  be  the  coordinates  of  the  centre  of 
inertia,  and  9ft  the  mass  of  the  entire  system  ;  then,  for 
continuous  forces, 


Wise  =  2w  -j-r  =  2X     I 
dt 


Wly  =  2»» 


df 

d-z 


S7 


3tts  = 

*maW 

Also, 

for 

impulses, 

m  {u  -  u) 

=  2w(m 

m{v  -v) 

=  2m  (v 

Wl  {re  -  of) 

=  2»i(?c 

=  2Z 


(19) 


ii )  =  2X  \ 

tO=2F,l.  (20) 

w)  =  2Z  ) 

These  equations  give  the  motion  of  the  centre  of  inertia 
of  a  free  system  acted  on  by  any  forces.  From  them  it 
appears  that — 

The  centre  of  inertia  of  a  free  system  moves  as  if  all  the  forces 
were  applied  to  the  entire  mass  concentrated  there. 

206.  Constraints  and  Partial  Freedom.  — If  a  system 
be  subject  to  external  constraints,  we  may  apply  equations 
(19)  and  (20),  provided  we  suppose  the  constraints  replaced 
by  the  forces  to  which  they  give  rise. 

If  a  system,  though  not  entirely  free,  be  such  that  equal 
and  parallel  displacements  of  arbitrary  magnitude  can  be 
given  to  each  of  its  points  in  a  definite  direction,  and  if  the 
axis  of  x  be  taken  in  that  direction,  we  have  still  the  equation 

m  =  2X 

R 


242  The  General  Dynamical  Principles. 

207.  Internal  Forces. — Any  force  by  which  two 
parts  of  a  system  act  on  each  other  is  said  to  be  internal. 

Since  action  and  reaction  are  eqnal  and  opposite,  the 
components  of  internal  forces  destroy  one  another  in  the  sums 
EX,  E  F,  and  S^.     Hence  in  any  system 

Interna  I  forces,  whether  continuous  or  impulsive,  have  no  effect 
on  the  motion  of  the  centre  of  inertia. 

208.  Case  of  no  External  Forces. — It  follows  from 
Art.  207,  that  if  a  system  be  acted  on  by  no  external  forces, 
its  centre  of  inertia  is  either  at  rest  or  moves  in  a  straight 
line  with  a  constant  velocity.  This  theorem  is  sometimes 
termed  The  Principle  of  the  Conservation  of  the  Motion  of  the 
Centre  of  Inertia. 

Kesults  similar  to  those  of  the  preceding  Articles  hold 
good  for  impulses. 

209.  Motion  of  a  Free  System  relative  to  its 
Centre  of  Inertia. — If  £,  rj,  Z  be  the  coordinates  of  any 
point  of  a  system  referred  to  axes  through  its  centre  of 
inertia  parallel  to  fixed  directions,  x  =  x  +  £,  y  =  y  +  rj,  s  =  z  +  Z- 
Substituting  in  D'Alembert's  equation,  we  have 

*(ax-  (»,)  j£)+*(ar-  <*0  §}  »(**-(*-)  g) 


o. 


.,((x-5f)«*(r-5)**(,-5)^ 

Now  2w£  =  Emij  =  Sm£  =  0 ;  hence,  by  equations  (19), 
Art.  205,  we  obtain 

It  follows  from  this  equation  that — 
The  motion  of  a  free  system  relative  tojts  centre  of  inertia  is 
the  same  as  if  this  point  were  fixed  in  space,  the  applied  forces 


Moments  of  Momentum.  243 

being  unaltered  as  regards  magnitude,  direction,  and  point  of 
application. 

The  theorem  just  stated  holds  good  as  well  for  impulsive 
as  for  continuous  forces.  This  readily  appears  by  applying 
the  transformation  employed  above  to  equation  (4),  Art.  197, 
and  making  use  of  equations  (20),  Art.  205. 

210.  Moments  of  Momentum. — It  is  readily  seen  that 
at  any  instant  the  expression 

2m  (xv  -  yu)  or  2m  (xy  -  yx) 

represents  the  entire  moment  of  the  momenta  round  the  axis 
of  z  of  all  the  elements  of  the  system  at  the  instant :  and 
similarly  2m  (yz  -  zy)  and  2m  (zx  -  xz)  represent  the  corre- 
sponding quantities  relative  to  the  axes  of  x  and  y,  respec- 
tively. 

These  moments  of  momenta  are  of  fundamental  impor- 
tance in  the  discussion  of  the  motion  of  any  system,  and  we 
shall  accordingly  represent  them  by  distinct  symbols. 

Thus  let 

Sx  =  2m  [yz  -  zy),  S2  =  2m  (zx  -  xz),  S3  =  2m  {xy  -  yx),  (22) 

then  equations  (18)  may  be  written  in  the  form  J 

S1-Sl'=L,    H2-H:=M,    Hz-Hs'  =  N,       (23) 

in  which  IZi',  Si,  Si  represent  the  moments  of  momenta  of 
the  system  before,  and  Slf  S2,  Sz  those  after,  the  impact. 

If  the  body  be  at  rest  when  acted  on  by  the  impulses,  these 
equations  become 

SX  =  L,     S2  =  M,     S,  =  N.  (24) 

Hence,  in  this  case,  the  moments  of  momenta  generated 
by  the  impulses  are  respectively  equal  to  the  impulsive 
moments  applied. 

Next,  since  xy  -yx  =  —  (xy  -  yx), 

dSx 

we  have  —  =  2m  (xy  -  yx), 

r  2 


244  The  General  Dynamical  Principles. 

and  it  follows  that  equations  (16)  may  be  expressed  in  the 
following  form : — 

^-  =  i'  ^r=Jf'  ^=if-  (25) 

The  quantities  Hlf  H2,  H3  admit  of  an  important  trans- 
formation, as  follows  : — 

If  \  It 3  dt  represent  the  elementary  area  described  round  the 
origin  by  the  projection  of  the  point  xyz  on  the  plane  of  xy, 
then 

xy  -  yx  =  hz. 

Hence  H*  =  S;w/i3,  and,  likewise,  representing  the  pro- 
jections on  the  planes  of  yz  and  xz  by  a  similar  notation, 

Hi  =  SffiAi,     H2  =  "2,mh2. 

Accordingly  equations  (25)  may  be  written 

Sm-rr=i:,      Sm-^  =  M,     ^m-^  =  N.  (26 

dtf  atf  ^ 

The  corresponding  equations  for  impulses  are 

2mA,  =  X,     2w/i2  =  if,     2mA3  =  JV".  (27) 

If  the  system  is  in  motion  when  the  impulses  act,  the  three 
latter  equations  should  be  written 

^mhi  =  L  +  ^m/h\  ^mh2  =  M+  %mk2',    *2mhz  =  N+  2mA3',  (28) 

where  hf,  h2,  h{  are  the  values  of  hi,  h2,  h3  the  instant  before 
the  impulses  act. 

The  quantities  hiy  h2,  h3,  &c,  are  the  areal  velocities, 
relative  to  the  origin,  of  the  different  points  of  the  system ; 

and  —,  &c,  are  the  areal  accelerations  (see  Art.  29). 
av 

In  any  system  in  motion  the  three  moments  Hh  H2,  Hz, 

if   they  were   regarded   as   moments   of   forces   or   couples 

acting  on  the  same  rigid  body,  would  be  equivalent  to  a 


Moments  of  Momentum.  245 

single   moment    H  round   a   line   whose   direction    cosines 

TT  TT  TT 

are  ~,     — -2,     -=^ ;    H  being  given  by  the  equation 

a     a     n 

E2  =  H?  +  Hi  +  -ff33. 

This  line  is  called  the  momentum  axis  of  the  system  relative 
to  the  origin.  As  it  is  the  axis  of  the  couple  which  is  the 
resultant  of  the  couples  corresponding  to  the  moments  of  the 
momenta  of  the  different  elements  of  the  system,  it  is  plain 
that  its  direction  is  independent  of  the  directions  of  the  co- 
ordinate axes. 

If  Sdt  be  twice  the  sum  of  the  projections  of  the 
elementary  areas  described  by  all  the  points  of  the  system 
round  the  origin,  each  multiplied  by  the  corresponding 
element  of  mass,  on  a  plane  whose  normal  makes  an  angle 
6  with  the  momentum  axis,  then 

S  =  Hcos  6.  (29) 

This  may  be  proved  in  the  following  manner :  Let  hdt 
be  double  the  elementary  area  described  by  the  element 
whose  mass  is  m  round  the  origin ;  and  let  a,  /3,  y  be  the 
cosines  of  the  angles  its  plane  makes  with  the  coordinate 
planes;  then,  A,  /u,  v  being  the  direction  cosines  of  the  normal 
to  the  plane  of  8, 

S  =  ,2mh  (a\  +  (5fi  +  yv)  =  X^mha  +  ju2w/*|3  +  v2mhy 
=  A^1  +  ju^2  +  v5-3  =  ^JAj  +  /xJ2  +  vJ3j  =  ^cos0. 

Hence,  the  multiple  sum  of  the  projections  of  the  elementary 
areas  on  the  plane  at  right  angles  to  the  momentum  axis  is  a 
maximum. 

This  plane  is  called  the  Principal  Plane  relative  to  the 
origin.  From  what  has  been  just  proved,  we  see  again 
that  its  position  is  independent  of  the  directions  of  the  axes. 

If  g,  n,  Z  be  a  second  set  of  rectangular  axes  through  the 
origin  parallel  to  directions  fixed  in  space,  and  if  the  direc- 


246  The  General  Dynamical  Principles. 

tion  cosines  of  £,  referred  to  x,  y,  z,  be  ah  a2,  a3 ;  of  r\,  bly  b2,  b3 ; 
of  £,  Ci,  c2,  c3 ;  we  have,  as  particular  cases  of  what  has  been 
proved  above, 

2m(ij£  -  £rj)  =  Exax  +  E2a2  +  E:ia3  \ 

2m(Z%  -  &)  =  ExbL  +  E2b2  +  Ezh      .         (30) 

2m  (ty  -  r)%)  =  Excx  +  E2c2  +  EzCi  ) 

The  preceding  theorems  of  this  Article  are  true  for  any 
system  of  moving  points,  and  whether  the  origin  be  fixed 
or  movable. 

Again,  to  find  the  moments  of  momentum  of  a  system  round 
axes  intersecting  at  a  point  whose  coordinates  are  a,  b,  c. 

Let  Ei,  E2,  E3  be  the  moments  of  momentum  of  the 
system  round  axes  parallel  to  the  coordinate  axes,  and  inter- 
secting at  the  point  abc ;  then,  we  have 

m=-2m{[y-b)i-(z-c)y) 

=  2m  (yz  -  zy)  -  (b^mz  -  c'Emy) ; 

but  x,  y,  z  being  the  coordinates  of  the  centre  of  inertia  of 
the  system, 

2my  =  my,     Sms=3ffs; 
hence  we  obtain 

E;=Ex-m(bz-~cT/)  \ 

e:  =  E2-m{cx-  ai)  [  •  (31) 

E3'=E3-Wl(ay-bic)  ) 

Again,  the  moment  of  momentum  of  a  system  round  an  axis, 
through  any  point  0,  is  equal  to  the  moment  of  the  momentum 
relative  to  the  centre  of  inertia  round  a  parallel  axis  through 
that  point,  together  with  the  moment  of  momentum  round  the  axis 
through  0  of  the  entire  mass  of  the  system  supposed  to  be  con- 
centrated at  the  centre  of  inertia,  and  moving  with  it. 


Internal  Forces.  247 

Take  the  axis  through  0  as  the  axis  of  %,  and  make  use 
of  the  transformation  employed  in  Art.  209,  then 

2m  (yz  -  zy)  =  Sm  { {[/  +  n)  (k  +  £)  -  (z  +  Z)  (y  +  y) ) 

=  m(fi-zf/)  +  2m(vt-Zv);  (32) 

since  2mr}  =  HmZ,  =  0,     Smrj  =  Sm§  =  0. 

The  student  will  observe  that  £,  17,  &o.,  denote  relative,  not 
absolute,  velocities.  If  the  origin  0  be  the  centre  of  inertia 
of  the  system,  equations  (23),  (24),  and  (25)  hold  good  whether 
0  be  fixed  or  moving  (Art.  209),  the  axes  being  parallel  to 
lines  fixed  in  space. 

In  the  deduction  of  equations  (23),  (24),  and  (25),  we 
have  supposed  that  the  system  is  free,  that  is,  unacted  on  by 
constraints  external  to  the  system  itself. 

211.  Constraints  and  Partial  Freedom. — A  system 
which  is  not  free  may  be  regarded  as  free,  if  the  external 
constraints  be  replaced  by  the  forces  to  which  they  give  rise. 

In  general,  we  can  ascertain  whether  a  given  constraint 
affects  the  validity  of  equations  (23),  (24),  and  (25),  by  con- 
sidering its  influence  on  the  conditions  of  equilibrium  of  a 
rigid  body. 

If  one  point  of  a  rigid  body  be  fixed,  we  know  that  for 
its  equilibrium  the  moments  of  the  applied  forces  round  three 
rectangular  axes  meeting  at  the  point  must  each  be  equal  to 
zero.     Hence  we  conclude — 

If  there  be  one  point  of  a  system  fixed,  equations  (23),  (24), 
and  (25)  hold  good  for  this  point  as  origin. 

Again,  if  there  be  a  fixed  line  in  a  rigid  body,  the  con- 
dition of  equilibrium  is  that  the  moment  of  the  applied 
forces  round  this  line  should  be  zero.     From  this  we  infer — 

If  there  be  a  fixed  line  in  a  system,  the  rate  of  change  relative 
to  the  time  in  the  moment  of  momentum  of  the  system  round  this 
line  is  equal  to  the  moment  of  the  applied  forces. 

212.  Internal  Forces. — Since  internal  forces  occur  in 
pairs,  each  pair  consisting  of  two  equal  and  oj)posite  forces 


248  The  General  Dynamical  Principles. 

having  a  common  line  of  direction,  the  moment  round  any 
line  of  the  whole  set  of  internal  forces  must  be  zero.  Hence 
the  moments  of  momentum  of  any  system  are  unaffected  by  forces 
internal  to  the  system. 

213.  Conservation  of  Moment  of  Momentum. — If 
a  free  system  be  unacted  on  by  any  forces  external  to  itself, 
its  resultant  moment  of  momentum,  relative  to  any  point 
fixed  in  space,  is  constant,  and  has  for  its  axis  a  line  whose 
direction  is  invariable. 

A  similar  result  holds  good  for  the  centre  of  inertia  even 
though  this  point  be  not  fixed  in  space. 

If  a  system,  otherwise  free,  contain  a  point  or  a  line  fixed 
in  space,  and  be  unacted  on  by  external  forces,  the  resultant 
moment  of  momentum  of  the  sj^stem  relative  to  the  fixed 
point,  or  the  moment  of  momentum  round  the  fixed  line,  is 
constant. 

The  theorems  enunciated  in  this  Article  together  consti- 
tute what  has  been  often  termed  The  Principle  of  the  Conser- 
vation of  the  Moment  of  Momentum,  or  The  Principle  of  the 
Conservation  of  Areas. 

As  the  moment  of  a  force  round  an  axis  intersecting  the 
line  of  direction  of  the  force  is  zero,  we  see  that — 

If  the  lines  of  direction  of  all  the  external  forces  which  act  on 
a  free  system  be  met  by  the  same  space  axis,  the  moment  of  mo- 
mentum of  the  system  round  this  axis  is  constant. 

If  the  space-axis  be  fixed  in  the  system,  which  is  other- 
wise free,  the  theorem  above  still  holds  good. 

In  a  similar  manner  we  may  conclude  that — 

If  a  system  receive  an  impulse,  the  moment  of  momentum-  of 
the  system  round  an  axis  fixed  in  space,  and  passing  through  any 
point  on  the  line  of  direction  of  the  impulse,  remains  the  same  as 
before. 

Examples. 

1.  In  any  system  in  motion,  show  that  the  moments  of  momenta  round 
three  rectangular  axes  are  equal  to  the  moments  of  the  impulses  which  would 
impart  to  the  system  if  at  rest  its  actual  motion. 

2.  If  |,  ?j,  £  he  the  coordinates,  relative  to  the  centre  of  inertia,  of  any  point 
of  a  free  system,  show  directly,  that  if  the  system  start  from  rest, 

»»(^'-^)  =  2(^-fr),  &c, 


Examples.  249 

and  that  during  the  motion, 

Ey  equation  (32),  Art.  210,  we  have 

but  2fty  =  2F,         and        27*2  =  2/?;         therefore,  &c. 

Again,  differentiating  each  side  of  the  equation  (32)  of  Art.  210,  we  have 
m{y't-  zy)  +  2m{r,C-  Cv)  =  Zm{y~-zy)  =  Z  =  2{  (y +  rj)  Z-  (»  +  {)Y}  ; 

and  as  Wty  =  2F,     m'z  =  ^Z, 

we  obtain  the  required  result. 

3.  A  satellite  of  mass  m  is  moving  in  a  circle  whose  radius  is  r,  round  a 
planet  whose  mass  is  M,  and  which  rotates  round  an  axis  perpendicular  to  the 
plane  of  the  orbit  with  an  angular  velocity  n.  If  Cbe  the  moment  of  inertia 
of  the  planet,  and  /x  the  attraction  between  unit  masses  at  the  unit  of  distance, 
show  that  the  moment  of  momentum  of  the  system  round  its  centre  of  inertia  is 


cln  +  p^{M+m)-±f*\. 


4.  A  heavy  particle  moves  on  a  smooth  surface  of  revolution  whose  axis  is 
vertical ;  prove  that  the  moment  of  momentum  of  the  particle  round  the  axis 
is  constant. 

5.  A  number  of  mutually  attracting  particles  are  acted  on  by  forces  passing 
through  the  same  fixed  point ;  prove  that  their  resultant  moment  of  momentum 
relative  to  this  point  is  constant,  and  that  the  direction  of  its  axis  is  invariable. 

6.  A  system  is  acted  on  by  no  external  force  except  gravity  ;  prove  that  its 
moments  of  momenta  round  axes  parallel  to  fixed  directions  in  space,  and  inter- 
secting at  its  centre  of  inertia,  are  constant. 

7.  Show  that  the  centre  of  inertia  of  the  universe  is  either  fixed  in  space  or 
else  moves  in  a  straight  line  with  a  constant  velocity. 

8.  A  man  walks  from  one  end  to  the  other  of  a  uniform  plank  which  is 
placed  on  a  smooth  horizontal  table  ;  determine  the  displacement  of  the  plank. 

Let  a  be  the  length  of  the  plank,  P  its  mass,  M  that  of  the  man  ;  the  dis- 

placement  is  — a. 

9.  A  uniform  plank  is  placed  on  a  smooth  inclined  plane,  so  as  to  be  perpen- 
dicular to  the  intersection  of  the  inclined  plane  with  the  horizon ;  determine  the 


250  The  General  Dynamical  Principles. 

time  in  which  a  man  should  go  from  the  upper  to  the  lower  end  of  the  plank  in 
order  that  it  should  remain  unmoved. 

Let  t  he  the  time  required.     The  displacement  of  the  centre  of  inertia  of  the 

M 
system  in  the  time  t  in  space  is  \gt2  sin  i,  and  relative  to  the  plank  is  — — -a. 

If  the  plank  remain  unmoved  these  must  he  equal.     Hence 
f       2M         a 


M+  F  g  sini 

10.  The  base  of  a  smooth  homogeneous  circular  semi-cylinder  rests  on  a  hori- 
zontal plane.  A  particle  m  is  placed  at  a  point  on  the  surface  of  the  semi- 
cylinder,  situated  in  a  vertical  plane  containing  its  centre  of  inertia  and  perpen- 
dicular to  its  axis.     Show  that  the  particle  will  describe  an  ellipse. 

Let  the  axis  of  x  be  the  intersection  of  the  vertical  plane,  in  which  the 
particle  moves,  with  the  horizontal  plane  on  which  the  semi-cylinder  rests  ;  the 
axis  of  y  being  vertical.  Let  x,  y  be  the  coordinates  of  the  particle,  x'  the  co- 
ordinate of  the  centre  of  inertia  of  the  semi -cylinder,  m  its  mass,  and  a  its 
radius. 

Considering  the  whole  system  as  one  body,  we  have  (Art.  206), 

d2x  ,  d2x' 

m  — —  +  m  — -  =  0. 
dt1  dt2 

Hence,  since  the  system  starts  from  rest,  mx  +  m'x  is  constant,  or  the  pro- 
jection on  the  horizontal  plane  of  the  centre  of  inertia  of  the  whole  system 
remains  fixed  in  space.     Taking  this  point  for  origin,  we  have  mx  +  m'x'  =  0. 

Again,  since  the  semi-cylinder  is  homogeneous,  we  have,  from  the  geometri- 
cal conditions, 

(x  —  x')2  +  y2  =  a2. 

Substituting  for  x',  we  obtain 

(m  +  m')2  x2  +  m'2y2  =  m'2a2. 

11.  Two  particles,  connected  by  a  rigid  rod  whose  weight  is  negligible,  are 
projected  along  a  smooth  horizontal  plane  ;   determine  their  motion. 

The  position  of  the  centre  of  inertia  at  any  time  is  given  by  the  equations 

x  =  mt  +  a,     y=nt+  b, 

and  the  inclination  of  the  rod  to  the  axis  of  x  by  the  equation  9  =  tot  +  e,  where 
m,  n,  a,  b,  w,  and  e  are  constants. 

12.  Two  equal  particles  are  connected  together  by  a  fine  inextensible  string  ; 
one  of  them  is  placed  on  a  smooth  table,  the  other  just  over  the  edge,  the  string 
being  at  fnll  stretch  at  right  angles  to  the  edge  ;  find  the  interval  of  time  from 
the  instant  at  which  the  particle  originally  on  the  table  leaves  it  to  the  instant 
at  which  the  string  first  becomes  horizontal. 

The  acceleration  of  the  particle  moving  on  the  table  is  \g.  Hence,  if  c  be 
the  length  of  the  string,  the  particle  leaves  the  table  with  a  horizontal  velo- 
city v,  where  v2  =  gc.  At  this  instant  the  middle  point  of  the  string  has  a 
horizontal  velocity  \v,  and  the  lower  particle  has  no  horizontal  velocity.     Hence 


Examples.  251 

the  moment  of  momentum  of  the  system  round  a  horizontal  axis  through  the 
centre  of  inertia  is  \mcv.  This  remains  constant  (Ex.  6),  and  therefore  twice 
the  area  described  round  the  centre  of  inertia  in  any  time  t  is  \mcvt.  If  t  be  the 
interval  of  time  during  which  the  string  passes  from  a  vertical  to  a  horizontal 
position,  we  have,  therefore,  J?rc2  =  \cvt,  and  substituting  for  v  its  value,  we 
obtain 


-*£ 


13.  A  sphere  is  projected  with  a  velocity  v  along  a  uniform  smooth  tube 
within  which  it  fits  exactly.  The  tube  rests  on  a  smooth  horizontal  plane,  and 
its  axis  forms  a  circle ;   determine  the  motion. 

Let  m  be  the  mass  of  the  sphere,  ml  that  of  the  tube,  and  a  the  radius  of 
the  circle  formed  by  its  axis.  The  common  centre  of  inertia  0  of  the  tube 
and  sphere  moves  parallel  to  the  direction  of  projection  of  the  sphere   with 

a  velocity  ,  and  the  centres  of  the  tube  and  sphere  describe  circles  round 

m  +  m 

v 
0  with  an  angular  velocity  -. 

14.  A  spherical  shell  rests  upon  a  smooth  horizontal  plane;  a  particle  is 
placed  at  the  lowest  point  of  the  internal  surface  of  the  shell,  which  is  then 
projected  with  a  horizontal  velocity  V.  The  internal  surface  of  the  shell  being 
smooth,  determine  to  what  height  the  particle  will  ascend. 

Let  x  and  y  be  the  coordinates  of  the  particle,  m  its  mass,  and  v  its  velocity  ; 
x'  and  y  the  coordinates,  and  v  the  velocity  of  the  centre  of  the  shell,  m  being 
its  mass.  Take  as  axis  of  x  the  intersection  of  the  smooth  horizontal  plane  with 
the  vertical  plane  of  motion ;  then,  Art.  200, 

mvz  +  m'v'2  =  m'V2  -  2mgy, 
and,  by  Art.  206, 

mx  +  mx'  =  m'  V. 

Also,  as  the  particle  remains  on  the  sphere  whose  radius  is  a,  we  have 

(x-x')2  +  {y  -y'f  =  ai; 

whence,  differentiating,  and  remembering  that  y'  =  0,  we  have  x  -  x'  =  0  when 
y  =  0.     Hence,  substituting,  we  obtain 

m       r2 


2  (m  +  m')   g 

This  result  may  not  hold  good  if  the  value  of  y  given  above  exceed  a. 

15.  A  smooth  tube,  movable  in  a  horizontal  plane  about  a  vertical  axis,  is 
charged  with  a  number  of  balls  at  given  intervals ;  an  angular  velocity  Q.  is 
communicated  to  the  tube  ;  determine  the  velocities  of  the  tube  and  of  the 
balls  at  any  assigned  distances  of  the  latter  from  the  axis. 

Let  mi,  mo,  &c.  be  the  masses  of  the  balls,  a\,  a2  &c.  their  initial  distances 


252  The  General  Dynamical  Principles. 

from  the  axis,  n,  r2,  &c.  their  distances  at  any  instant,  w  the  angular  velocity, 
and  Mk2  the  moment  of  inertia  of  the  tube  about  the  axis  ;  then  (Arts.  213,  200), 

(mi  n2  +  m%  r22  +  &c.  +  Mk%)  w  =  (mi  «r  +  m%  a%-  +  &c.  +  Mk2)  n, 

mi  h2  4-  ^2  h2  +  &c.  +  (mi  rr  +  m%  r£  +  &c.  +  Mk2)  a?  =  (m\  a{*  +  mia£ 

4-  &c.  +  Mk2)  n2. 

(f2  Ti  d2  ?*o 

Again  (Art.  28),  —  -  n  «2  =  0  =  — —  r2  a>2, 

c?2n  d2r2 

whence  r2  —7-: —  n  -=-r-  =  0, 


and  integrating, 
Hence  we  have 


dt2  dt2 

dri  dr% 

r2  — n  — -  =  constant  =  0. 

dt  dt 


r\  «i 

—  =  constant  =  — , 


and  therefore  also   —  =  — ,  with  similar  equations  for  the  other  distances  and 
r2      a2 

velocities.    Substituting  in  the  equations  of  momentum  and  vis  viva,  and  putting 

mi  «i2  4-  mz  «22  +  &c.  =  7,     Mk2  =  I', 
we  obtain 

(7?-i2  +  7'«i2)  a?  =  (7+  7')  «i3  a,  (ln2  +  r  ar)  h2  =  (7+  7')  «i*(na  -  «i2)  n2,  &c. 

16.  An  indefinitely  great  number  of  thin  cylindrical  shells  are  revolving 
in  the  same  direction  about  their  common  axis,  the  angular  velocity  of  each  shell 
being  proportional  to  a  positive  power  of  its  radius.  If  the  system  of  shells  be 
suddenly  united  into  a  solid  cylinder,  find  the  angular  velocity  of  the  cylinder 
about  its  axis. 

Let  w  be  the  angular  velocity  of  any  shell,  r  its  radius,  fl  and  R  being  those 
of  the  outermost  shell,  then  w  =  Ar»,  and  before  the  shells  are  united,  the  moment 

2ttA       r»+3  dr. 
Jo 


of  momentum  of  the  system  is 


If  to'  be  the  angular  velocity  of  the  united  svstem,  its  moment  of  momentum  is 
fi  — —  w  .     Equating  these  two,  we  obtain 

a)  = -. 

n  +  4 

17.  A  uniform  horizontal  stick  falling  to  the  ground  strikes  at  one  end 
against  a  stone  ;  compare  the  blow  it  receives  with  what  it  would  have  received 
had  both  ends  struck  simultaneously  against  two  stones,  the  blows  being  sup- 
posed to  be  perpendicular  to  the  stick. 


Examples.  253 

Let  v  and  v  be  the  velocities  of  the  middle  point  C  of  the  stick,  before  and 
after  it  receives  the  single  blow  at  the  extremity  A  ;  let  za  be  the  length  of 
the  stick,  m  its  mass,  and  P  the  impulse  of  the  blow.  The  moment  of  momentum 
of  the  stick  round  a  horizontal  space  axis  through  A  remains  unaltered  by  the 
blow.  Before  the  blow  the  whole  moment  of  momentum  is  due  ( (32),  Art.  210) 
to  the  motion  of  the  centre  of  inertia,  the  stick  having  no  motion  relative  to  it. 
After  the  blow  the  stick  is  rotating  round  A  (since  this  point  is  reduced  to  rest) 
with  an  angular  velocity  ».  Hence  f  ma2  <a  =  mav';  but  v  =  aw,  and  therefore, 
substituting,  we  have 

v  =  %v',     and     v'  -  v  -\v'. 

Again,  from  the  motion  of  the  centre  of  inertia  C,  we  obtain 
P  =  m  [v  —  v')  =  —  i  mv'. 

In  the  second  case,  when  the  stick  receives  two  blows  each  equal  to  Q,  it  is 
reduced  to  rest,  and  therefore 

2Q  =  -  mv',     or     Q  =  -  J  mv'. 

Therefore  finally  P=±Q- 

If  the  stick  be  elastic,  the  above  investigation  holds  good  for  the  impulses 
received  during  the  first  period  of  each  impact ;  and  as  the  total  impulses  are  in 
a  constant  ratio  to  the  former,  the  result  is  unaffected  by  the  elasticity  of  the 
stick. 


254     ) 


CHAPTER  X. 

MOTION    OF    A    RIGID    BODY    PARALLEL    TO    A   FIXED    PLANE. 

Section  I. — Kinematics. 

214.  Rigid    Body,   Determination  of  its  Position. — 

A  body  is  said  to  be  rigid  when  its  constitution  is  such  that 
the  relative  position  of  its  points  with  respect  to  each  other  is 
unalterable. 

The  position  of  a  point  in  space  is  usually  determined  by 
means  of  three  rectangular  coordinates,  and  depends  therefore 
upon  three  independent  quantities.  It  is  easy  to  see  that  the 
position  of  a  rigid  body  is  determined  by  six  independent 
variables.  For,  the  position  in  space  of  a  definite  point  A  of 
the  body  is  determined  by  three  independent  variables  ;  two 
more  are  required  to  determine  the  plane  in  space  in  which  a 
definite  plane  a  of  the  body  passing  through  A  should  lie ; 
and  finally,  one  more  is  necessary  to  fix  in  this  plane  a  definite 
line  of  the  body  passing  through  A,  and  lying  in  the  plane  a. 
When  the  position  in  space  of  every  point  of  the  plane  a  is 
determined,  it  is  obvious  that  the  positions  of  all  points  of 
the  rigid  body  are  completely  determined,  since  perpendiculars 
from  them  on  the  plane  a  are  invariable  in  magnitude. 

215.  Degrees  of  Freedom. — As  six  independent 
quantities  are  required  to  determine  the  position  of  a  rigid 
body,  such  a  body,  if  subject  to  no  restraint,  is  said  to  have 
six  degrees  of  freedom. 

It  is  plain,  from  what  has  been  said,  that  if  the  positions 
of  three  points  of  a  rigid  body  not  lying  on  the  same  straight 
line  are  fixed,  the  position  of  every  point  of  the  body  is  de- 
termined. 

216.  Motion  of  Translation. — When  a  body  moves 
so  that  the  elements  of  the  paths  described  by  its  different 
points  are  equal  and  parallel  straight  lines,  the  motion  is  said 
to  be  one  of  translation. 


Motion  of  Rotation.  255 

The  path  described  by  any  one  point  of  a  body  is,  in 
general,  a  curve,  and  it  appears  from  the  above  definition 
that  the  curves  described  by  the  different  points  during  any 
motion  of  translation  are  equal  and  similar.     Hence — 

In  a  motion  of  translation,  the  line  joining  any  two  definite 
points  of  the  body  remains  parallel  to  its  initial  position. 

As  the  distances  traversed  by  each  point  of  the  body  are 
the  same  both  in  magnitude  and  direction,  we  may  speak  of 
the  motion  of  translation  of  the  body,  and  may  compound  any 
number  of  elementary  motions  in  the  same  manner  as  for  a 
point. 

217.  Motion  of  Rotation. — As  already  stated  in  Art/ 
95,  when  a  body  is  moving  in  such  a  manner  that  each  point 
is  describing  the  arc  of  a  circle  having  its  centre  on  a  fixed 
straight  line,  to  which  its  plane  is  perpendicular,  the  motion 
is  said  to  be  a  rotation,  and  the  fixed  straight  line  passing 
through  the  centres  of  all  the  circles  is  called  the  axis  of 
rotation. 

In  a  motion  of  this  kind  every  point  of  the  body  lying 
on  the  axis  of  rotation  remains  fixed  during  the  motion. 

All  lines  in  the  body  perpendicular  to  the  axis  of  ro- 
tation turn  through  the  same  angle,  which  is  called  the 
angular  rotation,  or  simply  the  rotation  of  the  body. 

Any  line  AB  of  the  body  which  lies  in  a  plane  at  right  angles- 
to  the  axis  of  rotation  makes,  at  the  end  of  the  motion,  an  angle 
with  its  initial  position,  which  is  equal  to  the  angular  rotation  of 
the  body. 

This  readily  appears  as  follows  : — Join  A  to  the  point  O 
in  which  the  axis  meets  the  plane  in  which  AB  lies  ;  then, 
A  and  B'  being  the  new  positions  taken  by  A  and  B,  since 
OA  makes  the  same  angle  with  AB  which  OA'  makes  with 
AB',  the  quadrilateral  formed  by  OA,  OA,  AB  and  AB 
can  be  inscribed  in  a  circle,  and  therefore  the  angle  between 
AB'  and  AB  is  equal  to  that  between  OA  and  OA. 

It  is  easy  to  see,  that  if  two  positions  of  a  body  have  a 
straight  line  of  particles  in  common,  the  body  can  be  moved 
from  one  of  these  positions  to  the  other  by  a  rotation  round 
this  line. 


256  Kinematics  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane. 

218.  Motion  Parallel  to  a  Fixed  Plane. — When 
the  paths  described  by  the  several  points  of  a  body  during  its 
motion  are  made  up  of  elements,  each  of  which  is  parallel  to 
the  same  fixed  plane,  the  motion  of  the  body  is  said  to  be 
parallel  to  this  plane. 

If  we  consider  any  definite  plane  section  of  the  body, 
which  at  the  beginning  of  the  motion  is  parallel  to  the  fixed 
plane,  this  section  must  continue  in  the  same  plane  through- 
out the  motion,  and  its  position  at  each  instant  determines 
the  position  of  every  point  of  the  body.  In  order,  therefore, 
to  study  the  motion  of  a  body  moving  parallel  to  a  fixed 
plane,  we  have  merely  to  investigate  the  motion  of  a  plane 
figure  in  its  own  plane. 

219.  Motion  of  a  Plane  Figure  in  its  own  Plane. 
— A  plane  figure  can  be  moved  from  any  one  position  in  its 
own  plane  to  any  other  by  giving  it  first  a  motion  of  trans- 
lation, whereby  any  arbitrary  point  A  of  the  figure  is  moved 
from  its  first  position  AY  in  space  to  its  second  position  A2, 
and  then  a  motion  of  rotation  round  a  perpendicular  axis 
passing  through  A2,  whereby  a  definite  line  AB  of  the  figure 
is  moved  into  its  final  position  in  space  A2B2.  As  the  point 
A  is  perfectly  arbitrary,  the  motion  may  be  effected  in  an 
infinite  variety  of  ways.  The  motion  of  translation  to  be 
given  to  the  figure  differs  in  general  according  as  different 
points  of  the  figure  are  chosen  for  A,  but  the  motion  of  rotation 
is  in  all  cases  the  same. 

This  readily  appears  from  Art.  217,  as  the  initial  and  final 
positions  of  any  definite  line  of  the  figure  are  given,  and  the 
angle  between  them  is  in  all  cases  the  rotation  of  the  body. 

The  results  arrived  at  above  depend  upon  the  fact  that 
the  position  of  a  plane  figure  in  its  own  plane  is  completely 
determined  by  the  position  of  one  definite  straight  line  of  the 
figure.  Hence  also  it  appears  that  by  properly  selecting  the 
point  A,  the  motion  of  translation  may  in  general  be  dis- 
pensed with  altogether,  or,  in  other  words  {Differential  Cal- 
culus, Art.  293) — 

A  plane  figure  can  be  moved  from  any  one  position  into 
any  other  in  its  oivn  plane  by  a  rotation  round  a  point  fixed 
in  the  plane. 


Composition  of  Velocities.  257 

In  fact,  BC  being  the  original  position  of  any  definite  line 
of  the  figure,  and  B'C  its  new  position;  if  we  join  BB\ 
bisect  it,  and  erect  a  perpendicular,  and  do  the  same  with 
CC\  these  two  perpendiculars  will,  in  general,  determine 
by  their  intersection  a  point  0,  a  rotation  round  which  effects 
the  given  change  of  position. 

If  BB'  be  parallel  to  CC  this  construction  fails.  Two 
cases  then  arise,  according  as  BB'  is  equal  to  CC  or  not. 
In  the  latter  case,  the  intersection  of  BC  and  B'C  is  the 
centre  of  rotation.  In  the  former  the  motion  is  one  of  pure 
translation,  and  the  point  0  is  at  infinity. 

As  a  particular  case,  it  follows  that — 

Two  rotations  effected  successively  round  two  parallel  axes 
bring  a  body  into  the  same  position  as  a  single  rotation  round  an 
axis  parallel  to  the  two  former,  the  single  rotation  being  equal  in 
magnitude  to  the  sum  of  the  two  to  which  it  is  equivalent. 

We  see  also  that — 

A  rotation  round  any  given  axis  brings  a  body  into  the  same 
position  as  an  equal  rotation  round ra  parallel  axis  through  any 
arbitrary  point,  together  with  a  motion  of  translation. 

Hence  it  appears  that — 

Equal  and  opposite  rotations  effected  successively  round  two 
parallel  axes  A  andB  are  equivalent  to  a  single  motion  of  trans- 
lation. 

For,  a  rotation  round  A  is  equivalent  to  an  equal  rotation 
round  B,  together  with  a  translation ;  therefore  equal  and 
opposite  rotations  round  A  and  B  are  equivalent  to  equal 
and  opposite  rotations  round  B,  together  with  a  translation  ; 
but  equal  and  opposite  rotations  round  B  destroy  each  other ; 
therefore,  &c. 

220.  Composition  of  Velocities. — Hitherto  we  have 
been  considering  displacements  having  a  finite  magnitude. 
In  regard  to  such  displacements  the  order  in  which  the  several 
motions  are  effected  is  of  importance,  and  in  order  to  arrive 
at  definite  results  it  is  necessary  to  specify  whether  the 
successive  axes  of  rotation  are  fixed  in  space  or  in  the  body. 
In  Kinetics,  we  are  for  the  most  part  concerned  not  only  with 
the  initial  and  final  positions  of  a  body,  but  also  with  the 


258     Kinematics  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane. 

mode  in  which  it  passes  from  the  one  position  to  the  other. 
It  becomes  then  necessary  to  consider  the  infinitely  small 
motions  through  which  the  body  assumes  its  successive  posi- 
tions. Displacements  effected  in  the  same  element  of  time 
divided  by  that  element  then  become  velocities,  and  the  com- 
position and  equivalence  of  infinitely  small  displacements  is 
tantamount  to  the  composition  and  equivalence  of  velocities. 

If  the  displacements  received  by  a  body  be  infinitely  small,  it 
is  indifferent  in  what  order  rotations  are  effected  round  different 
axes  fixed  in  space. 

For,  the  changes  produced  in  the  coordinates  of  any  point 
of  the  body  by  such  a  rotation  are  functions  of  its  amplitude, 
and  of  the  initial  values  of  those  coordinates.  In  the  present 
case  these  changes  are  infinitely  small,  and  therefore  altera- 
tions in  them,  due  to  a  previous  displacement  which  is  itself 
infinitely  small,  are  infinitely  small  quantities  of  the  second 
order. 

Again,  from  similar  considerations,  it  appears  that  it  is 
indifferent  whether  the  axes  be  fixed  in  space  or  axes  fixed  in  the 
body,  tvhose  positions  at  the  commencement  of  the  infinitely  small 
motion  coincide  with  those  of  the  axes  fixed  in  space. 

When  the  order  of  two  displacements  is  indifferent  they 
may  be  regarded  as  simultaneous,  and  if  the  resultant  dis- 
placement be  such  as  to  move  the  body  from  one  position  into 
the  next  successive  position,  it  is  the  actual  motion  of  the  body. 

221.  Motion  of  a  Rigid  Body. — The  theorems  of 
Article  219,  when  applied  to  infinitely  small  motions  of  a 
rigid  body  parallel  to  a  fixed  plane,  give  the  following 
results : — 

(1).  The  motion  of  a  body  parallel  to  a  fixed  plane  con- 
sists at  any  instant  of  a  velocity  of  rotation  w  round  an  axis, 
passing  through  any  arbitrary  point  A  of  the  body,  at  right 
angles  to  the  plane,  and  a  velocity  of  translation  v  parallel  to 
the  plane. 

(2).  At  each  instant  there  is  an  axis,  called  the  instan- 
taneous axis  of  rotation,  which  is  such  that  a  velocity  of 
rotation  w  round  it  represents  the  whole  motion  of  the  body. 

If  r  be  the  distance  from  A  to  this  axis,  and  v  the  velocity 
of  translation  which  the  body  must  be  considered  to  possess 


Rotations  round  Parallel  Axes.  259 

when  the  axis  of  rotation  is  regarded  as  passing  through  A, 
then,  in  order  that  the  axis  should  be  at  rest  at  the  instant,  the 
direction  of  v  must  be  at  right  angles  to  r,  and  we  must  have 
r  =  rtt).  We  thus  see,  that  in  a  rigid  body  a  motion  of  rota- 
tion together  with  a  motion  of  translation,  in  a  direction 
perpendicular  to  the  axis  of  rotation,  can  be  compounded  into 
a  motion  which  is  one  of  rotation  solely.  Also,  such  motions 
cannot  be  compounded  into  a  single  rotation  unless  the 
direction  of  the  translation  is  perpendicular  to  the  axis  of 
rotation. 

(3) .  Two  coexisting  velocities  of  rotation  round  parallel 
axes  are  equivalent  to  a  single  velocity  of  rotation,  equal  to 
their  sum,  round  an  axis  parallel  to  the  two  former,  and  di- 
viding the  distance  between  them  inversely  as  the  component 
velocities. 

(4).  Two  equal  and  opposite  velocities  of  rotation  whose 
common  magnitude  is  w,  round  parallel  axes,  are  equivalent 
to  a  velocity  of  translation,  perpendicular  to  the  plane  of 
the  axes,  whose  magnitude  is  aw,  where  a  is  the  distance  be- 
tween the  axes. 

We  see  from  what  is  stated  above  that  velocities  of  rota- 
tion round  parallel  axes  are  compounded  like  parallel  forces. 

In  considering  rotations  round  parallel  axes  it  is  necessary 
to  take  into  account  not  only  the  magnitudes  of  the  rotations, 
but  also  their  algebraical  signs,  or  directions.  The  axis  of  rota- 
tion is  supposed  to  be  drawn  from  the  feet  of  the  spectator  to 
his  head,  so  that  in  estimating  rotations  the  axis  points  towards 
the  spectator.  If  two  rotations  round  parallel  axes,  viewed 
from  the  same  side  of  the  plane  perpendicular  to  the  axes,  are 
both  in  the  same  direction,  they  have  like  algebraical  signs. 
The  positive  direction  of  rotation  is  of  course  arbitrary  ;  but  in 
the  application  of  Analytic  Greometry  to  rotational  displace- 
ments it  is  usual  to  regard  rotations  as  positive  which  bring 
a  point — from  the  axis  of  X  positive  to  the  axis  of  Y  positive, 
from  Y  positive  to  Z  positive,  and  from  Z  positive  to  X  posi- 
tive. It  follows  from  this,  that  if  the  axes  of  X  and  Y  be 
drawn  in  the  usual  manner,  a  rotation  opposite  in  direction 
to  that  of  the  hands  of  a  watch  is  to  be  regarded  as  positive 
(Art.  87).     Such  a  rotation  is  termed  counter-clockwise. 

s  2 


260     Kinematics  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane. 

222.  Analytical  Treatment  of  the  Motion  of  a 
Plane  Figure  in  its  own  Plane. — When  a  point  moves 
in  a  circle  whose  centre  is  the  origin,  we  may  assume 


x  =  r  cos  xp,     y  =  r  sin  \p, 

dx  .     ,  dip      dt/  .  dip 

whence  —  =  -  r  sm  \p  — ,     —  =  r  cos  xp  — , 

dt  dt       dt  dt 

d\L  dx  dy  . 

and  putting         —  =  w,     —  =  x ,     —  =  y,  we  nave 

x  =  -  put,     y  =  xw,  (1) 

for  the  rotation  of  the  point  xy  round  the  origin. 

Now  let  x',  y  be  the  coordinates  of  a  definite  point  of  a 
plane  figure  referred  to  axes  fixed  in  space  ;  x,  y  those  of  any 
point  of  the  figure  referred  to  the  same  axes ;  £,  rj  those  of 
the  same  point  of  the  figure  referred  to  axes  fixed  in  the  body 
and  meeting  at  the  point  xy:  moreover,  let  the  axis  of  £ 
make  the  angle  \p  with  the  axis  of  x  ;  then 

x  =  x  +  S  cos  \Jj  -  r)  sin  \P,     y  =  y  +  £  sin  ^  +  r)  cos  tf>, 

x  =  x  -  (£  sin  \)y  +  »j  cos  \p)  w 


(2) 
if  =  y  +  (?  cos  \fj  -  7]  sin  xp)  id 

Or, 

x  =  x  -  (y  -  y')w,     y  =  y  +  {x  -  x)  w.  (3) 

These  equations  show  that  the  velocity  of  the  point  xy  is 
made  up  of  two  parts — one  a  velocity  of  translation,  the  other 
a  velocity  of  rotation,  as  in  (1),  round  an  axis  through  xy. 

For  any  other  definite  point,  x"y"  of  the  figure  we  have, 
in  like  manner, 

x  =  x"  -  {y  -  y")  J\    y  =  y"+(z-  x")  <»". 

Equating  these  values  of  i  and  y  to  the  former,  and  compar- 
ing the  results  with  the  equations 

x  =x  -  (y  -y  )to  ,    y  =y  +  [x  -  x  )  w  , 


Pure  Rolling.  261 

we  see  that  w"  =  w,  or  the  velocity  of  rotation  to  be  attributed 
to  the  body,  is  the  same  whatever  be  the  point  through  which 
the  axis  of  rotation  is  supposed  to  pass. 

223.  Instantaneous  Centre,  Body  Centrode,  Space 
Centrode. — If  we  put  x  =  0,  y  =  0  in  equations  (2),  we  get 
the  coordinates  of  the  instantaneous  centre  of  rotation,  referred 
to  axes  fixed  in  the  body.  In  like  manner  equations  (3)  give 
the  coordinates  of  the  same  point  referred  to  axes  fixed  in 
space.  If  we  call  the  coordinates  of  the  instantaneous  centre 
£oj  *?o ;  ffoj  i/o,  respectively,  we  have 

&>  =  -  {%  sin  \p  -y  cos  \p),  7j0  =  -  (x  cos  \p  +  y'  sin  \p),    (4) 

W  (t) 

,   1  ./         ,  1  ./ 

x0  =  x  -  -  y,   y0  =  y  +-x.  (5) 


If  x,  y,  (i),  and  \p  are  known  functions  of  the  time  t, 
we  can  find  from  equations  (4),  by  eliminating  t,  the  path 
described  in  the  body  by  the  instantaneous  centre. 

From  equations  (5)  we  can  find  in  the  same  manner  the 
path  described  by  the  instantaneous  centre  in  space. 

The  former  of  these  curves  is  called  the  body  centrode,  the 
latter  the  space  centrode. 

The  student  must  carefully  distinguish  between  the  in- 
stantaneous centre  and  the  point  of  the  body  which  coincides 
with  it  at  any  instant.  The  latter  has  no  velocity  at  the 
instant  either  in  space  or  in  the  body  ;  the  former  (the 
instantaneous  centre)  has  in  general  a  velocity  both  in  space 
and  in  the  body. 

224.  Pure  Rolling. — In  pure  rolling  the  points  of  one 
curve  or  surface  come  into  contact  successively  with  those  of 
another,  the  relative  tangential  velocity  of  the  points  of  con- 
tact being  zero.  If  one  curve  or  surface  be  fixed  in  space, 
the  motion  of  the  other  consists  of  a  series  of  rotations  round 
axes  through  the  successive  points  of  contact  (Differential 
Calculus,  Art.  295).  In  the  case  of  one  plane  curve  rolling 
on  another,  this  appears  as  follows  :  — 


262     Kinematies  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane. 

Let  QQ'  be  the  curve  fixed  in  space,  and  PPf  the  one 
which  rolls  on  it,  P,  Pr  being 
two  consecutive  points  on  the 
latter.     By  hypothesis,  P  has 
no  velocity  along  the  tangent  at  P 

Q,  and  at  the  end  of  an  infi-  ^~o^ 

nitely  short  time  P'  coincides 
with  Q\  and  the  distance  between  P  and  Q  is  then  an  infi- 
nitely small  quantity  of  the  second  order.  Hence,  while 
other  points  of  the  body  have  received  infinitely  small  dis- 
placements of  the  first  order,  P  has  received  one  of  the 
second  order,  and  has,  therefore,  no  velocity  in  any  direc- 
tion. Hence,  during  the  instant  under  consideration,  the 
body  must  be  rotating  round  an  axis  through  P  (Art.  217). 
It  is  obvious  that  the  acceleration  of  P  in  the  direction  of  the 
tangent  at  Q  is  zero ;  and  it  can  be  easily  seen  that  its  acce- 
leration in  the  direction  of  the  normal  is  in  general  finite, 
and  equal  in  magnitude  to  Uto,  where  w  is  the  angular  velo- 
city of  the  body,  and  U  is  the  velocity  of  the  instantaneous 
centre  of  rotation,  this  point  having  moved  during  the  instant 
from  Q  to  Q'  in  space,  and  from  P  to  P'  in  the  body. 

225.  Geometrical  Representation  of  the  Motion 
of  a  Body  moving  Parallel  to  a  Fixed  Plane. — When 
a  body  is  moving  parallel  to  a  fixed  plane,  if  we  can  deter- 
mine the  space  centrode  and  the  body  centrode,  the  motion  of 
the  body  is  completely  determined,  as  it  consists  of  the 
rolling,  without  slipping,  of  the  body  centrode  on  the  space 
centrode. 

The  geometrical  applications  of  the  principles  laid  down 
in  the  present  and  preceding  Articles  are  numerous  and  im- 
portant ;  but  as  they  do  not  fall  within  the  scope  of  the 
present  treatise,  the  reader  is  referred  for  them  to  Chap.  xix. 
of  the  Differential  Calculus,  and  to  Minchin's  Uniplanar  Kine- 
matics,  Chap.  in. 

226.  Velocity  of  any  Given  Point  of  a  Body. — In 

Kinetics  the  motion  of  a  body  is  usually  regarded  as  made  up 
of  a  motion  of  translation  v,  and  a  motion  of  rotation  w,  round 
an  axis  through  the  centre  of  inertia  G.  It  is  sometimes 
important  to  determine  the  velocity  of  a  given  point  A  of  the 


Examples.  263 

body.     In  the  case  of  motion  parallel  to  a  fixed  plane  this  is 
readily  done  analytically  by  equations  (3). 

Otherwise,  geometrically : — let p  be  the  distance  from  A  to 
the  axis  of  rotation  through  G,  then,  owing  to  the  rotation, 
A  has  a  velocity  pco  perpendicular  to  the  plane  passing 
through  A  and  the  axis  of  rotation,  and  this,  combined  with 
the  velocity  of  translation  v,  gives  the  velocity  of  A. 

Examples. 

1.  Show  directly  that  if  a  body  have  two  equal  and  opposite  velocities  of 
rotation  round  two  parallel  axes,  the  velocity  of  any  point  is  at  right  angles  to 
the  plane  containing  the  parallel  axes,  and  is  equal  to  the  distance  between  the 
axes  multiplied  by  the  angular  velocity. 

Draw  a  plane  through  the  point  at  right  angles  to  the  two  parallel  axes. 
Describe  round  the  axes  circles  passing  through  the  point.  The  component 
velocities  of  the  point  are  perpendicular  and  proportional  to  the  radii  of  these 
circles,  and  the  resultant  velocity  is,  therefore,  in  the  direction  of  the  common 
chord,  and  proportional  to  the  line  joining  the  centres. 

2.  Prove  that  a  velocity  of  rotation  round  any  axis  is  equivalent  to  an  equal 
velocity  of  rotation  »  round  a  parallel  axis,  together  with  a  velocity  of  transla- 
tion wa  along  a  line  at  right  angles  to  the  plane  containing  the  axes,  the  distance 
between  which  is  a. 

3.  A  body  receives,  in  a  given  order,  finite  rotations  round  two  parallel 
axes  fixed  in  space.  Determine  the  magnitude  of  the  equivalent  rotation,  and 
the  position  of  its  axis. 

4.  If  the  parallel  axes  round  which  the  body  receives  successive  rotations  be 
fixed  not  in  space  but  in  the  body,  determine  the  single  rotation  which  would 
bring  the  body  into  the  same  position. 

If  A,  B  are  the  intersections  of  the  nxesjixed  in  space,  with  a  plane  at  right 


angles,  £  that  of  the  resultant  axis,  and  o,  £,  x,  the  magnitudes  of  the  rotations 


264     Kinematics  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane. 

round  them,  then  BAR  =  — f  a,  ABR  =  §  #,  and  the  resultant  rotation  x  =  a  +  £, 
or,  (a  -  /3),  according  as  a  and  jSare  in  the  same  or  opposite  directions.  In  the 
latter  case  its  direction  is  the  same  as  the  greater  of  the  two.  If  A  and  B'  are 
the  positions  of  the  axes  fixed  in  the  hody,  B'AR  =  ^  a,  AB'R  —  —  ^  $. 

5.  Two  equal  and  opposite  finite  rotations  round  parallel  axes  hring  a  body 
into  the  same  position  as  a  single  motion  of  translation.  Determine  the  direc- 
tion and  magnitude  of  this  translation. 

The  direction  of  translation  is  at  right  angles  to  a  line  which  makes  with 
AB  or  AB'  an  angle  equal  to  -  J  a,  or  J  o,  and  the  magnitude  of  the  translation 
=  1AB  sin  |  a,  or,  2AB'  sin  \  a. 

6.  If  the  direction  of  the  motion  of  each  point  of  a  hody  be  always  parallel 
to  a  fixed  plane,  the  motion  is  equivalent  to  a  succession  of  rotations  round  the 
generating  lines  of  a  cylinder  fixed  in  space,  which  is  at  right  angles  to  the  fixed 
plane. 

7.  A  plane  area  receives  a  motion  of  translation  in  its  own  plane  whose  com- 
ponents, parallel  to  the  axes,  are  a  and  b  ;  and  a  rotation  6  round  the  point  in  the 
body  which,  at  the  beginning  of  the  motion,  coincides  [with  the  fixed  origin. 
Determine  the  coordinates  of  the  point,  a  rotation  round  which  would  bring  the 
body  into  the  same  position. 

a  sin|0-  b  cos|0  b  sin  i0  +  a  cosi0 

Ans.  x  = — : — : =-,     y  = = s_. 

2sin£0  '     *  2sini0 

8.  Show  from  these  expressions  that  the  amplitude  of  the  rotation  is  the 
same  as  before. 


\a1  _j.    £2 

If  (p  be  the  amplitude,     sin  \  <p  =  |  J— =  sin^  9 ; 


9.  ABCD  is  a  frame  composed  of  three  bars  connected  by  joints  at  B  and 
C.  It  is  capable  of  moving  in  one  plane,  the  points  A  and  I)  being  fixed. 
Determine  the  relation  between  the  angular  velocities  of  AB  and  DC. 


At  any  instant  B  is  moving  in  a  circle  round  A,  or  at  right  angles  to  AB  ; 
and  C  at  right  angles  to  DC.  Hence  the  instantaneous  'centre  of  rotation  of  BO 
is  0,  the  point  of  intersection  of^4i?  and  CD  ;  wherefore  AB  .  cci  =  OB  .  a,  and 

DC .  &>2  =  OC.  o) ;  hence  —  =  -—  .  — —  (Thomson  a.xdTait). 

«2      AB     CO  - 

10.  A  bar  AB  moves  in  one  plane  with  given  angular  velocity  rounds, 
while  at  B  it  is  freely  jointed  to  another  bar  BC,  whose  extremity  Cis  con- 
strained to  move  along  a  fixed  straight  groove  passing  through  A  ;  find  the 
velocity  of  C  in  any  position. 


Examples. 


265 


Draw  a  perpendicular  to  AC  at  C,  and  let  it  meet  AB  in  0  ;  then  0  is  the 
instantaneous  centre  of  rotation  of  BC.     If  v  he  the  velocity  of  C,  and  w  the 

t9C 

angular  velocity  of  AB,  v  =  AB  .  —  .  a>  =  AP.  w,  where  ^P  is  drawn  at  right 

OB 
angles  to  AC  to  meet  PC  in  P.     For  the  further  discussion  of  this  question  the 
reader   is   referred   to   Minchin,    Uniplanar  Kinematics,    p.   47,    or  Goodeve, 
Elements  of  Mechanism,  Chap.  i.     The  arrangement  of  machinery  mentioned  in 
this  example  is  called  the  crank  and  connecting  rod. 

11.  A  har  moves  in  a  horizontal  plane  with  uniform  angular  velocity  round 
one  extremity.  To  the  other  extremity  a  horizontal  circle  is  attached.  If  the 
circle  he  regarded  as  rotating  round  its  centre,  what  additional  motion  must  it  be 
considered  to  have  ? 

A  velocity  of  translation  at  right  angles  to  the  har,  and  equal  to  aw,  where  a 
is  the  distance  of  the  centre  of  the  circle  from  the  fixed  end  of  the  har,  and  co  the 
angular  velocity. 

12.  If  two  definite  points  of  a  plane  figure  are  constrained  to  move  along 
two  straight  lines  in  its  plane,  which  are  fixed  in  space,  the  space  centrode  and 
the  body  centrode  are  circles,  the  former  being  double  the  latter  {Differential 
Calculus,  Art.  295). 

13.  In  Peaucellier's  arrangement  find  the  relation  between  the  velocity  of 
the  point  describing  the  straight  line  and  that  of  one  of  the  adjacent  corners  of 
the  parallelogram. 

M.  Peaucellier,  in  1864,  first  succeeded  in  transforming  circular  into  recti- 
linear motion  by  the  following  arrangement : — A  and  B  are  fixed  points ;  AP 
and  AQ  are  two  equal  bars  which  can  turn  freely  round  A  ;  BR  is  another  bar 
turning  freely  round  P,  and  equal  in  length  'to  AB ;  QRPX  is  a  jointed 
parallelogram  composed  of  four  equal  bars  turning  freely  round  their  points  of 
intersection.  If  a  motion  be  imparted  to  the  system,  the  points  P,  Q,  P  describe 
circles.     That  the  point  X  describes  a  straight  line  may  be  shown  as  follows : — 

In  any  position  of  the  system,  since  L  PRX  =  L  QRX,  and  L  PEA  =  L  QRA, 
XR  and  RA  are  in  one  straight  line ;  then  XPR  being  an  isosceles  triangle,  and 


PA  a  line  drawn  from  the  vertex  to  the  base,  AR .  AX  =  AP*-RP2  =  const. ; 
wherefore  X  describes  a  curve  which  is  the  inverse,  with  respect  to  A  as  origin, 


266     Kinematics  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane. 


of  that  described  by\R.     Now  the  point  R  describes  a  circle  which  passes  through 
A  ;  hence  X  describes  a  straight  line,  perpendicular  to  AB  at  the  point  8,  where 

AS.AD  =  AP*  -  RPK 

We  proceed  to  find  the  relation  between  the  velocities  of  P  and  X.  Draw 
XO  at  right  angles  to  SX  ;  then  0  is  the  instantaneous  centre  of  rotation  of  the 
bar  PX. 


Let  AP  =  a,  PX  =  b,  BR  (in  former  figure)  =  c ;  then  «  being  the  angular 
velocity  of  AP,  a'  that  of  PX,  and  v  the  velocity  of  X ;  we  have,  since  0  is  the 
instantaneous  centre, 

v  =  OX  .  «',  and  OP .  io' =  AP .  w  ; 
therefore 


=  °*.AP. 

OP 


AT.  w. 
ur 

Again,  if 

PAT=B,  PTA  =  cp,  we  have  AT  =  a  sin  6  (cot  6  +  cot  <p) 
therefore 

v  =  a  sin  Q  (cot  0  +  cot  (£>)  o>, 

where  (p  is  given  by  the  equation 


a  cos  6  +  b  cos 


2c 


14.  A  plane  area  is  moving  in  its  own  plane  ;  determine  the  accelerations  of 
any  point  in  it  parallel  to  the  tangent  and  the  normal  to  the  space  centrode  at 
the  instantaneous  centre  of  rotation. 

Let  xo,  yo  be  the  coordinates  of  a  point  fixed  in  the  lamina,  |,  77  those  of 
any  point  in  it  referred  to  xo,  yo  as  origin,  and  to  axes  parallel  to  those  of  x,  y ; 
then 

d£  drj 

It 
w  being  the  angular  velocity  of  the  body ;  whence 

dx      dxo 


■    Tt=*°> 


dt        dt 


-7JW, 


dy      dy0 

dT  =  -dT+^ 


Examples.  267 

d-x      d2  %o        o        du> 
d2y      d2y0      dot 

—  =    A t    —   CD"  7). 

Call  the  centrode  fixed  in  space  C,  that  fixed  in  the  body,  r.  The  velocity 
of  the  point  0  of  the  body  which  coincides  at  any  instant  with  the  instantaneous 
centre  of  rotation  is  zero.  At  the  next  instant  the  instantaneous  centre  of  rota- 
tion has  moved  to  the  consecutive  position  on  each  of  the  curves  C  and  r.  At 
the  end  of  this  instant  Ohas  a  velocity  in  the  normal  to  C  equal  to  FIco,  where  I,  T 
are  consecutive  positions  of  the  instantaneous  centre  on  the  tangent  to  C.  Hence 
the  acceleration  of  0  along  the  tangent  to  C  is  zero,  and  along  the  normal  to  C 

is  w2    *   if  we  put  /' /=  dff,  and  w  =  — .     Xow  if  p  and  p  be  the  radii  of  cur- 
dO  dt 

1       l       l     -.  •         -i  a  4. dd       1 

vature  of  C  and  I\  and,  if  we  put =  -,  it  is  easily  seen  that  —  =  -. 

p       p      jk  aa      it 

Hence,  if  xqi/q  coincide  with  0,  and  we  take  as  axes  the  tangent  and  normal 

to  C,  we  have 


d2x 
dt2  = 

—  co'- 

1 

doi 

~d7v 

d2y 
dt2 

oP~R 

+ 

dca 

~di*~ 

0T7J. 


15.  Determine  the  points  of  the  body  which  have  at  any  instant  (I)  no 
acceleration  parallel  to  the  tangent  to  C  at  the  instantaneous  centre  of  rotation  ; 
(2)  no  acceleration  parallel  to  the  normal. 

These  points  consist  of  two  straight  lines  in  the  body  at  right  angles  to  each 
other,  the  first  of  which  passes  through  the  instantaneous  centre  of  rotation. 

16.  Determine  at  any  instant  the  position  of  the  point  in  the  body  having 
no  acceleration. 

It  is  the  intersection  of  the  two  lines  mentioned  in  the  last  example. 

If  a  be  the  angle  which  the  line  of  non-tangential  acceleration  (Ex.  15) 
makes  with  the  axis  of  x,  the  coordinates  of  this  point  may  be  expressed  in  the 
form 

|  =  R  sin  a  cos  a,     77  =  R  sin2  a. 

These  expressions  readily  follow  from  the  equations  of  Ex.  14.     This  point  is 
called  the  acceleration-centre. 

17.  The  acceleration  of  any  point  of  the  body  is  the  same  as  if  the  body  were 
turning  round  the  acceleration-centre  as  an  absolutely  fixed  point. 

18.  All  points  of  the  body  which  have  a  common  acceleration  lie  on  a  circle 
having  the  acceleration-centre  as  centre. 

19.  Find  the  points  of  the  body  for  which  the  acceleration  normal  to  the 
path  described  by  the  point  is  zero. 

Take  the  centre  of  rotation  as  origin  of  £77 ;  any  point  is  describing  a  circle 
round  it ;  hence  the  line  joining  the  origin  to  £tj  is  the  normal  to  the  path  of 


268     Kinematics  of  Rigid  Body  Moving  Parallel  to  Fixed  Plane, 

the  latter;  and  if  N  be  the  normal  acceleration,  and  r  the  distance  from  the  in- 
stantaneous centre  of  rotation, 


'-H-^-ff')+r(-"  +  ?«-")- 


£2  +  7T  7? 

•r f-  -  a/  it. 


r  r 

Hence,  at  any  instant,  the  points  for  which  N  =  0  lie  on  the  circle 

e  +  t  =  R-n- 

This  circle  passes  through  the  instantaneous  centre  of  rotation,  touches  the 
curve  C,  and  has  a  radius  =  Ji?.  For  the  reason  stated  in  Ex.  21  it  is  called 
the  circle  of  inflexions. — Differential  Calculus,  Art.  290. 

20.  Determine  the  points  of  the  system  for  which  the  acceleration  along  the 
path  is  zero. 

They  lie  on  a  circle  whose  equation,  referred  to  the  centre  of  rotation  as 
origin,  is 

and  which  passes  through  the  instantaneous  centre  of  rotation  and  cuts  the 
curve  C  orthogonally. 

The  theorems  of  the  last  two  examples  are  due  to  Bresse  {Journal  de  Vecole 
poly  technique,  t.  xx.). 

21.  Determine  at  any  instant  the  points  of  the  body  which  are  passing  over 
points  of  inflexion  on  their  respective  paths. 

v2  . 
They  are  the  points  having  no  normal  acceleration  (Ex.  19) ;  for,  as       is 

then  zero,  and  v  not  zero,  p  must  he  infinite. 

22.  Determine  the  coordinates  of  the  acceleration  centre  referred — (1)  to  axes 
fixed  in  space  ;  (2)  to  axes  fixed  in  the  body  (see  Article  223). 

Let  x\,  y\,  |i,  t?i  be  the  coordinates  in  question,  then,  %' ,  y'  being  the  space- 
coordinates  of  the  point  of  intersection  of  the  body-axes,  we  have 

{co2  +  co4}  [x\  —  x')  =  -  co  y'  +  oj2  x', 

{or  +  co4}  (yi  —y')  =  obx'  +  co2 y, 

{or  +  co4}  |i  =  co  (if  sin  i//  -  y*  cos  ij/)  +  a2  (x"  cos  if/  -f  y'  snuj/), 

{c62  +  co4}  ?ji  =  ci  (if  cos  -if  +  y  sin  \p)  —  co2  (x  sin  \p  -  y  cos  \p). 

Section  II. — Kinetics. — Constrained  Motion. 

227.     Special     Cases    of    Motion.        Degrees     of 

Freedom. — In  order  to  transform  the  general  equations  of 
motion  in  such  a  way  as  to  be  of  use  in  particular  problems, 
it  is  necessary  to  know  something  of  the  special  conditions  of 
the  problem  which  it  is  required  to  solve. 

We  have  seen  in  Article  214  that  six  conditions  are  re- 
quired to  fix  the  position  of  a  rigid  body,  and  we  have  found 


Kinetics,  Constrained  Motion.  269 

accordingly  six  equations  of  motion  for  a  body  perfectly  free. 
Such  a  body  is  said  to  have  six  degrees  of  freedom  (Art.  215). 
We  have  obtained  the  equations  for  this  case  in  their  most 
general  form  (Art.  204),  but  we  shall  now  adopt  the  reverse 
method  of  procedure,  and  consider  the  special  equations  to  be 
employed  for  a  body  having  one  degree  of  freedom. 

228.  One  Degree  of  Freedom. — A  body  is  said  to 
have  one  degree  of  freedom  when  its  position  is  limited  in 
such  a  way  as  to  depend  on  a  single  indeterminate  quantity. 
It  will  be  shown  subsequently  that  the  variations  of  the  co- 
ordinates of  any  point  of  a  body  entirely  free  are  linear  func- 
tions of  six  undetermined  quantities.  If  these  six  quantities 
are  connected  together  in  such  a  way  that  one  being  given 
all  the  rest  are  determined,  the  body  has  one  degree  of 
freedom. 

The  simplest  cases  of  one  degree  of  freedom  occur  when 
some  of  the  six  undetermined  displacements  are  zero.  We 
shall  consider  here  only  two  cases. 

(1).  If  the  motion  of  the  body  be  limited  to  a  series  of 
pure  translations,  and  the  path  of  one  of  its  points  be  as- 
signed. 

(2)  If  the  motion  of  the  body  be  limited  to  a  rotation 
round  an  axis  fixed  in  space. 

In  the  first  case  the  problem  is  readily  reducible  to  that 
of  the  constrained  motion  of  a  particle. 

This  reduction  is  most  easily  effected  by  employing 
D'Alembert's  Principle  as  expressed  by  Lagrange.  In  fact 
we  have 

*-4>"(r-'"S'K(*-'''SH-0- 

Now,  by  the  conditions  of  the  question,  $x,  c^  Sz  must  be 
the  same  for  every  point  of  the  body,  and  ds  being  the  arc 
of  the  curve  described  by  the  centre  of  inertia, 

ds  ds  ds 


270  Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

Making  these  substitutions,  we  obtain  the  single  equation  of 
motion, 

d%x\dx      (        d2i/\dy      /        d2z\dz 

=  (SX)|+(SF)|  +  (^)|; 

or,  as  dsz  =  da?  +  dy~  +  dz*, 

we  have  finally,  if  we  put  tyfl  for  the  whole  mass  of  the  body, 

3»S  -  a,  (i) 

where  #  is  the  sum  of  the  components  of  all  the  applied 
forces  along  the  tangent  to  the  path  of  the  centre  of  inertia ; 
but  this  is  obviously  the  equation  required  for  determining 
the  constrained  motion  of  a  particle. 

229.  Hotion  of  a  Body  round  an  Axis  fixed  in 
Space. — The  condition  of  equilibrium  of  a  rigid  body  having 
a  fixed  axis  is,  that  the  moment  of  the  forces  round  this  axis 
should  be  zero.  Take  the  fixed  axis  as  axis  of  x,  then  the 
single  equation  of  motion  is  the  first  of  equations  (18)  or 
(16),  Art.  204,  according  as  the  forces  acting  on  the  body 
are  impulsive  or  continuous.  Adopting  the  notation  of  Art. 
210,  the  equation  of  motion  is  thus  : 

Let  p  be  the  perpendicular  on  the  axis  from  any  point  P  of 
the  body,  a>  its  angular  velocity  at  any  instant,  and  /  its 
moment  of  inertia  round  the  axis;  then,  since  pu)  is  the 
velocity  of  the  particle  P,  its  moment  of  momentum  is  mp2w, 
and  Hy  =  wSmp2  =  Iw.  Substituting  this  value  for  Hi,  and 
remembering  that  /  is  constant,  we  obtain  as  the  equation  of 
motion  in  the  case  of  impulses 

J(w  -  w)  =  L,  (2) 


Examples.  271 

and  in  the  case  of  continuous  forces 

Equation  (3)  was  obtained  before  in  Art.  138  by  a  different 
method. 

230.  Equation  of  Vis  Viva  for  a  Body  moving 
round  a  Fixed  Axis.. — The  expression  for  the  vis  viva  of  a 
body  moving  round  a  fixed  axis  has  been  given  already, 
Art.  133.  If  we  take  the  fixed  axis  for  the  axis  of  a?,  we 
have,  as  the  equation  of  vis  viva, 

Lo2  =  22j(Fd>  +  Zdz)  +  c.  (4) 


Examples. 

1 .  To  the  ends  of  a  thin  light  piece  of  wood  are  fastened  spheres  of  lead 
•whose  masses  are  P  and  P'.  The  piece  of  wood  turns  on  a  horizontal  axis 
through  its  middle  point.  Its  length  being  21,  and  its  mass  negligible,  deter- 
mine the  time  of  a  small  oscillation,  the  spheres  being  so  small  that  the  squares 
of  their  radii  are  negligible  as  compared  with  /. 

A  \l       JP+  P' 

Ans. 


'^At= 


P' 


By  changing  P,  and  comparing  the  times  of  oscillation,  an  apparatus  of  the  kind 
mentioned  can  be  used  to  verify  the  Laws  of  Motion. 

2.  A  heavy  pendulum,  capable^  of  revolving  round  a  horizontal  axis,  is 
struck  when  at  rest  by  a  bullet  moving  in  a  horizontal  direction  at  right  angles 
to  the  fixed  axis.  The  bullet  remains  in  the  pendulum.  If  b  be  the  distance 
of  the  extremity  of  the  pendulum  from  the  axis,  c  the  distance  traversed  by  that 
extremity  under  the  influence  of  the  shot,  a  the  distance  from  the  axis  at 
which  the  bullet  penetrates,  v  the  velocity  of  the  bullet  at  impact,  m  its  mass, 
M  that  of  the  pendulum,  k  its  radius  of  gyration  round  the  fixed  axis,  and  p  the 
distance  of  the  latter  from  the  centre  of  inertia ;  prove  that 

v  =  —  ^J{g{MTc1  +  ma?)(Mp  +  ma)}. 

A  pendulum  such  as  that  described  above  is  called  a  Ballistic  Pendulum.  It 
has  been  employed  by  numerous  Physicists  to  determine  the  velocity  of  bullets. 

3.  A  plane  area  is  made  to  rotate  with  an  angular  velocity  w'  round  a  fixed 
axis  in  its  own  plane  by  the  expenditure  of  a  given  amount  of  work.  When 
rotating  it  strikes  a  sphere  of  mass  m,  at  a  distance  a  from  the  fixed  axis,  whose 


272     Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plan  e. 

velocity  at  the  instant  of  impact  is  zero.  Determine  the  moment  of  inertia  of 
the  plane  area  round  the  fixed  axis  in  order  that  the  velocity  imparted  to  the 
sphere  should  be  a  maximum. 

If  R  be  the  impulse  on  the  sphere  in  the  first  period  of  impact,  v  its  velocity, 
and  oj  the  angular  velocity  of  the  lamina  at  the  end  of  this  period, 

mv  =  R,     I(u)  -«')=—  aR,     aw  -  v, 

lalca' 


whence  R  = 


/+  ma2 


Tbe  whole  impulse  given  to  the  sphere  is  (1  +  e)R.     Hence  R  is  to  be  a  maxi- 

y/l 

mum ;  but  Iio'2  =  given  constant ;  therefore =  maximum  ;  and  therefore 

1+  ma1 

I  =  ma2. 

4.  In  Atwood's  machine,  if  the  pulley  be  not  perfectly  rougb,  and  slipping 
takes  place,  determine  the  motion  :  the  weight  of  the  rope  and  the  friction  of 
the  pulley  on  the  axle  being  neglected. 

If  an  acceleration  equal  and  opposite  to  that  by  which  it  is  actually  animated 
were  applied  to  each  element  of  the  string  it  would  be  in  equilibrium  ;  but  the 
mass  of  the  string  being  negligible,  the  force  corresponding  to  this  acceleration 
is  zero  g.p.  Hence  the  other  forces  acting  on  the  element  of  the  string  are  in 
equilibrium,  and  fj.  being  the  coefficient  of  friction,  and  T,  T'  the  tensions  of 
the  two  ends  of  the  rope  (Minchin,  Statics),  T  =  Te'^  =\T. 

If  z  be  the  height  from  the  ground  of  the  ascending  weight  W,  M  the  mass 
of  the  pulley,  A' its  radius  of  gyration,  a  its  radius,  0  the  angle  through  which  it 
has  turned,  we  have  also 


T  -  W" 

W      9 

_d2z 
"dt2 

W-T 
-     W9 

a(T- 

T). 

a2 
If  the  pulley  be  homogeneous,  k2  =  — ,  and  we  have  finally, 

a 


2 WW       d2z      KW -  W 
-9, 


\W+  W1     dt2      \W+  W" 


d-0      ..,        .  WW' 


dt2        v         'M{\W+  W'Y 

5.  Taking  into  account  the  friction  on  the  axle,  and  supposing  the  outside  of 
the  pulley  to  be  perfectly  rough,  and  the  inside  to  slip  on  the  axle,  determine 
the  motion. 

The  mass  of  the  string  being  neglected,  we  may,  as  in  the  last  example, 
regard  it  as  acted  on  by  a  system  of  forces  in  equilibrium.  Hence  (as  this 
equilibrium  would  not  be  disturbed  if  the  string  were  rigid)  the  tensions  ^and 
T  at  its  extremities  must  equilibrate  the  pressure  and  friction  exerted  by  the 
pulley  against  the  string ;  and,  conversely,  T  and  T  must  be  equivalent  to  the 


Moments  of  Momentum.  273 

pressure  aud  friction  exerted  by  the  string  against  the  pulley.  Hence  we  may 
consider  the  pulley  as  acted  on  by  the  forces  T,  T1,  and  its  own  weight ;  and  if 
jPbe  the  horizontal,  and  Q  the  vertical,  pressure  on  the  axle,  and  fx  the  coefficient 
of  friction,  since  the  centre  of  inertia  of  the  pulley  is  at  rest,  we  have  (Art.  206), 
P  =  fiQ,  Q  =  T  4  T'  +  Mg  —  /xP.  The  moment  of  the  couple  resulting  from 
the  friction  is  /x(P  +  Q)a,  where  o  is  the  radius  of  the  axle,  and  may  therefore 
be  written  in  the  form  &(T+  T  +  Mg),  where  (1  +  fj2)  0  =  fi(l  +  fi)  a. 

d29 
Substituting  for  the  equation     31k2  —  =  a(T-T')   of  Ex.  4, 


the  equation  Mk2  —  =  a(T-  T)  -  &(T+  T  +  Mg); 

nd  remembering  that  as  the  pulley  is  perfectly  rough,  a — -  =  — ,  we  obtain,  if 

j8  a2 

we  put  v  =  -  and  assume  that  k2  =  — , 
a  2 

(l+2u)Mg  +  4(l  +  u)W 
Mg  + 2(1  -v)W+2{l  +  v)W       s 

(l-2p)Mg  +  ±(l-v)W 
Mg+2(\-v)W+2{\-rv)W'        ' 

&z  _  (1  -  v)  W-  (1  +  v)  W  -  vMg 
dP  ~  (1  -  v)  JF+  (1  +  v)  W  +  \Mg*9' 

6.  If  the  pulley  be  not  perfectly  rough,  and  slipping  of  the  string  on  the 
pulley  takes  place,  determine  the  motion,  taking  into  account  the  friction  on  the 
axle,  and  supposing  the  inside  of  the  pulley  to  slip  as  before. 

In  this  case,  as  in  Ex.  4,  the  acceleration  of  the  weights  is  quite  independent 
of  the  mass  and  size  of  the  pulley,  and  we  have 

T=    ^WW  ffiz  _\W-  w 

\W+  W'  '     dt*      \W+  W'9' 

d29       U{\-v-A(\  +  v)WW> 


d€~        (        Mg(\W+W) 

231.  ^loiuents  of  Moineutuiu  of  Body  having  iixed 

Axis. — The  expression  for  the  moment  of  momentum  of  a 
rigid  body  round  an  axis  fixed  in  space  was  found  in  Art. 
229.  Adopting  the  notation  of  that  article,  we  shall  now,  by 
a  more  general  method,  obtain  expressions  for  the  moments 
of  momentum  round  each  of  the  three  coordinate  axes. 

T 


274     Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

We  have  (Art.  222),  since  the  body  is  supposed  to  be 
rotating  round  the  axis  of  x, 

x  =  o,     y  =  -sw,     s  =  ycv  ; 
whence  by  (22),  Art.  210, 

Hx  =  w2m  [y2  +  z2),     H2  =  -  to^mxy,     Hz  =  -  w2«s.     (5) 

Also,  by  differentiation,  and  substitution  of  their  values  for 
x,  y,  and  s,  we  obtain 

tfJ^        du 

-tt  =  -  it  Swa^  +  co2Zmxz,   I  .  (6) 

dt  dt  J 

If  the  axis  of  rotation  be  a  principal  axis  for  the  origin, 
equations  (5)  and  (6)  become  g 

where  ^4  is  the  moment  of  inertia  of  the  body  round  the 
fixed  axis. 

232.  Acceleration  of  any  Point  of  a  Body  having 
a  Fixed  Axis. — If  we  differentiate  the  expression  for  x,  y, 
and  z  given  in  Art.  231,  and  then  substitute  in  the  results 
thus  obtained  the  values  of  x,  y,  z  already  employed,  we  get 

x  =  0,     y  =  -  <bz  -  id2!/,     z  =  (by  -  to2z.  (8) 

233.  Stresses  on  the  Axis  of  Rotation. — We  have 
seen  that  D'Alembert's  Principle  furnishes  at  once  the  single 
equation  of  motion  which  is  required  to  determine  the  velo- 
city and  position  of  a  body  rotating  round  a  fixed  axis.  The 
same  principle  enables  us  to  write  down  the  equations  which 
are  required  to  determine  the  stresses  on  the  axis. 


Stresses  due  to  Impulses.  275 

In  order  to  determine  these  stresses,  we  may  regard  the 
body  as  compelled  to  rotate  round  the  fixed  axis  by  forces 
acting  on  the  body  at  any  two  points  on  the  axis.  The  body 
is  then  to  be  considered  free,  but  the  magnitude  of  the  forces 
replacing  the  constraints  is  such  as  to  compel  the  body  to 
rotate  round  the  given  axis. 

These  forces  are  obviously  equivalent  to  a  single  force 
passing  through  the  origin,  and  a  couple  whose  axis  is  per- 
pendicular to  the  fixed  axis. 

The  stresses  on  the  axis  are  equal  and  opposite  to  the 
forces  by  which  we  have  supposed  the  axis  to  be  replaced. 

234.  Stresses  due  to  Impulses . — In  this  case  we  shall 
suppose  the  stresses  to  consist  of  an  impulse  passing  through 
the  origin  whose  components  are  X0,  Y0,  Z0,  together  with 
an  impulsive  couple  whose  components  round  the  axes  of 
H  and  2  are  M0  and  JV0. 

If  we  suppose  the  body  to  start  from  rest,  equations  (17) 
and  (18),  Art.  204,  become,  by  Art.  231, 

2X-X0  =  2,mx  =  0, 

2  F-  Y0  =  %my  =  -  wSmss  =  -  wWlz,  [ ,  (9) 

%Z  -  ZQ  =  %mz  =     u)Zm?/=      wSfflf/ 

L  =  io%m  {if  +  s2)  =  Iw,  \ 
M -  M0  =  -  oy^rnxf/,      >  • 
N-  ATo  =  -  w%mxz       / 


(10) 


When  io  has  been  found  from  the  first  of  equations  (10)  the 
remaining  five  equations  determine  the  stresses. 

If  the  fixed  axis  be  a  principal  axis  at  the  origin  the  last 
two  equations  become 

M-Mo  =  0,     iV-iv"o  =  0. 

Hence,  if  a  body  having  a  fixed  axis,  which  is  a  principal 
axis  for  one  of  its  points,  be  set  in  motion  by  an  impulsive 
couple  whose  plane  is  perpendicular  to  the  axis,  there  is  no 
impulsive  stress  couple. 

t2 


276     Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

From  this  we  infer,  that  if  a  body,  having  a  fixed  point 
0,  be  acted  on  by  an  impulsive  couple  in  one  of  the  principal 
planes  at  0,  it  will  commence  to  turn  round  the  axis  perpen- 
dicular to  the  plane  of  the  impulsive  couple.  Again,  if  the 
body  be  acted  on  by  an  impulse  whose  line  of  direction  is 
situated  in  one  of  the  principal  planes  at  0,  it  will  commence 
to  turn  round  the  normal  to  this  plane. 

For  a  free  body,  likewise,  having  0  for  its  centre  of  inertia, 
these  results  are  true ;  but,  in  the  case  of  the  second,  the  body 
has  also  an  initial  motion  of  translation. 

If  the  body,  before  the  action  of  the  impulses  X,  &c, 
be  already  rotating  round  the  fixed  axis  with  an  angular 
velocity  a/,  equations  (9)  and  (10)  still  hold  good  in  their 
final  form,  provided  w  -  id'  be  substituted  for  w. 

If  we  suppose  the  origin  0  to  be  the  centre  of  suspension, 
or  point  in  which  the  fixed  axis  is  met  by  the  perpendicular 
p  from  the  centre  of  inertia  G,  and  take  the  axis  of  y 
to  coincide  with  this  line,  and  if  we  denote  the  sum  of  the 
components  of  the  applied  impulses  parallel  and  perpen- 
dicular to  OGhy  P  and  Q,  and  the  corresponding  impulsive 
stresses  by  _P0  and  Q0,  equations  (9)  become 

IX  -  X0  =  0'y     P  -  P0  =  0,     Q  -  Q0  =  Wpu.      (11  j 

235.  Centre  of  Percussion. — If  a  body  receive  a  blow 
which  makes  it  begin  to  rotate  round  a  fixed  axis,  without 
causing  any  impulsive  pressure  on  the  axis,  the  point  in  which 
the  direction  of  the  blow  intersects  the  plane  containing  the 
fixed  axis  and  the  centre  of  inertia  is  called  the  centre  of 
percussion.  In  order  that  such  a  point  should  exist,  both 
the  axis  and  the  line  of  direction  of  the  impulse  must  fulfil 
certain  conditions,  which  we  proceed  to  investigate. 

In  this  case,  the  fixed  axis  being,  as  before,  the  axis  of  x, 
we  have,  by  hypothesis,  X,  =  0,  Y0  =  0,  Z0  =0,  M0  =  0, 
N0  =  0.  If  we  denote  the  components  of  the  impulse  due  to 
the  blow  by  X,  P,  $  ;  and  the  components  of  the  impulsive 
couple  which  it  produces  by  X,  M,  N;  equations  (11)  and 
(10)  become 

X  =  0,     P  =  0,   Q  =  Sfeto,  |  ^  ^ 

L  =  Iw,  M  =  -  u&mxy,  N  =  -u^mxz 


Stress  on  Fixed  Axis  during  Rotation.  277 

Since  X  =  0,  aud  P  =  0,  the  centre  of  inertia  must  lie  in  a 
plane  through  the  fixed  axis,  at  right  angles  to  the  direction  of 
the  impulse. 

Again,  since  X  =  0,  the  direction  of  the  blow  may  be 
supposed  to  lie  in  the  plane  of  yz,  and  therefore  the  resulting 
couple  has  no  components  in  the  plane  of  zx  or  of  xy ; 
accordingly,  M  =  0  and  N  =  0.  Hence,  we  have  %nxy  =  0, 
and  %mx%  =  0 ;  consequently,  the  axis  of  rotation  must  be  a 
principal  axis  for  the  point  in  ichich  it  is  met  by  its  shortest 
distance  from  the  line  of  direction  of  the  impulse.  If,  now,  qbe 
the  distance  from  the  fixed  axis  of  the  line  of  action  of  the 
blow,  L  =  Qq,  and  therefore  tylpq  =  I. 

If  Wlk2  be  the  moment  of  inertia  of  the  body  round  an 
axis  through  its  centre  of  inertia  parallel  to  the  fixed  axis, 
/  =  Wl  (k~  +  p2).     (Integral  Calculus,  Chap.  X.) 

Hence  q  = — . 


Accordingly,  the  distance  of  the  centre  of  percussion  from  the 
fixed  axis  is  the  same  as  that  of  the  centre  of  oscillation.  (Art. 
'136.) 

Moreover,  if  £,  rj,  £  be  the  coordinates  of  any  point  rela- 
tively to  the  centre  of  inertia, 

jxzdm  =  Sfflxz  +  \%Zdm  ; 

hence,  if  the  axis  of  suspension  be  parallel  to  a  principal 
axis  through  the  centre  of  inertia,  x  =  0,  and  the  shortest 
distance  between  the  direction  of  the  blow  and  the  fixed  axis 
passes  through  the  centre  of  inertia,  and  the  centre  of  percussion 
coincides  with  the  centre  of  oscillation. 

236.  Stress  on  Fixed  Axis    during   Rotation. — In 

accordance  with  Art.  233,  and  following  the  analogy  of  Art. 
234,  we  shall  suppose  the  stress  at  any  instant  to  consist  of  a 
force  passing  through  the  origin,  whose  components  are  X0, 
F0,  and  Z0,  together  with  a  couple  whose  components  round 
the  axes  of  y  and  z  are  M0  and  N0. 


278     Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

In  this  case,  by  Arts.  231  and  232,  equations  (15)  and 
(16),  Art.  204,  become 

2X  -  X0  =  0,  -| 

27-7o  =  -»-^w2,   y  (13) 

%Z-Z»=    mydy-Wlztv2} 

-  6D%mxy  +  to2%mxz  =  M  -  M0>  X  (14) 

i 

-  6)%mxz  -  to'^mxy  =  N-  NQ  J 

If  the  axis  of  rotation  be  a  principal  axis  for  the  origin, 
the  last  two  equations  reduce  to  M  -  MQ  =  0,  JV  -  iVo  =  0. 

If  also  the  couple  resulting  from  the  applied  forces  be 
perpendicular  to  the  axis  of  rotation,  we  shall  have 

M0  =  0,  and  JST0  =  0. 

Accordingly,  in  this  case,  the  stress  couple  vanishes  when 
the  axis  of  rotation  is  a  principal  axis  for  the  origin. 

If  the  axis  of  rotation  pass  through  the  centre  of  inertia 
of  the  body,  we  have 

2X-X0  =  0,     2F-F0  =  0,     2Z-Z0  =  0. 

Accordingly,  if  a  body  be  rotating  round  a  principal  axis 
through  its  centre  of  inertia,  no  external  forces  being  supposed 
to  act,  there  is  no  stress  on  the  axis,  and  the  body  will  continue  to 
rotate  round  that  axis  with  a  uniform  angular  velocity. 

This  result  was  obtained  before  in  Article  98. 

If  we  make  the  same  hypotheses  as  those  at  the  end  of 
Art.  234,  and  adopt  a  similar  notation,  equations  (13) 
become 

%X-Xo=0,     P-P0  =  -mpto\     Q-Q0  =  Wpd>.    (15) 

These  equations  of  motion  of  the  centre  of  inertia  can  of 
course  be  obtained  directly  from  the  consideration  that  this 


Examples.  279 

point  is  describing  a  circle  round  the  origin  with  an  angular 
velocity  w. 

In  general,  w  and  to  can  be  determined  from  the  first 
of  equations  (14),  and  the  stresses  can  then  be  found  from 
the  remaining  equations  of  this  Article. 


Examples. 

1.  A  rigid  body  is  turning  round  a  fixed  axis  under  the  influence  of  a  couple, 
whose  axis  is  parallel  to  the  axis  of  rotation  :  what  condition  must  be  fulfilled  in 
order  that  the  axis  should  suffer  a  pressure  at  only  one  point  ?  (Schell,  Theorie 
der  Bewegung  und  der  Krufte.) 

The  axis  of  rotation  must  be  a  principal  axis  at  this  point.  The  pressure  is 
then  at  right  angles  to  the  axis. 

2.  If  the  pressure  at  the  fixed  point  vanishes,  what  further  condition  must 
be  fulfilled  ? 

The  point  must  be  the  centre  of  inertia. 

3.  If  a  homogeneous  cubical  mass  at  rest  receive  an  impulse,  determine  the 
resulting  motion. 

4.  A  body  starting  from  rest  turns  under  the  action  of  gravity  round  a  fixed 
horizontal  axis,  which  is  a  principal  axis  at  the  centre  of  suspension.  Find  the 
stress  on  the  axis. 

Take  the  centre  of  suspension  (Art.  136)  for  origin,  and  the  fixed  axis  for 
that  of  x. 

Let  6  be  the  angle  which  the  line  joining  the  origin  to  the  centre  of  inertia 
makes  at  any  instant  with  a  horizontal  line  perpendicular  to  the  fixed  axis,  then 

o)  =  — ,  and  the  axis  of  x  being  a  principal  axis  at  the  origin,  the  stress  couple 

dt 
is  zero.     Again,  m  being  the  mass  of  the  body,  L  =  mpg  cos  0,  and  therefore, 

dco  _  d-d  _  gp  cos  0  m 
dt  ~  df~  ~  k*+pz  ' 
whence,  by  integration, 


■©■ 


F+p- 


(sin  0- sin  a), 


where  a  is  the  initial  value  of  0. 

Finally,  _P=  nig  sin  0,  and  Q  =  mg  cos  0  ;  whence,  substituting  their  values 
f or  P,  Q,  w-,  and  a  in  equations  (15),  we  obtain 


-  sin  9  -  7,        ,  sin  o  [ ,   Qo=  mg  p— — -  cos  6. 


+P*     v   &+p*      y  ™     *  &+p 


5.  In  Ex.  3  find  the  position  of  the  body  in  which  the  stress  on  the  axis  is 
a  minimum. 


280     Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

From  the  expressions  for  Po  and  Qo>  we  obtain 

P02  +  Q02  -     **  g      {k*  +  2k2p~  sin  6 (3  sin  6 -  2  sin  a)  +  pl  (3  sin  9  - 2  sin  a)'- } , 

and,  since  9  is  never  less  than  a,  this  expression  is  a  minimum  when  6  =  a. 

6.  A  bar,  revolving  with  an  angular  velocity  n  round  a  fixed  axis  perpendi- 
cular to  its  length,  strikes  perpendicularly  against  a  fixed  obstacle ;  find  the 
impulses  against  the  obstacle  and  the  axis,  and  the  angular  velocity  of  the  bar, 
after  collision. 

Let  0  be  the  point  in  which  the  fixed  axis  meets  the  bar,  G  its  centre  of 
inertia,  A  the  point  at  which  it  strikes  the  obstacle,  m  its  mass,  and  h  its  radius 
of  gyration  round  an  axis  through  G  parallel  to  the  fixed  axis  ;  let  R'  and  Q'  be 
the  magnitudes  of  the  impulses  produced  by  the  obstacle  and  the  axis  in  the 
first  period  of  impact,  R"  and  Q"  those  produced  in  the  second  period,  and  ca  the 
angular  velocity  after  collision  ;  then,  if  OG  =  a,  GA=  b,  since  the  velocity  of 
the  bar  is  reduced  to  zero  in  the  first  period,  we  have 

R'  +  Q'  =  man,     R'(a  +  b)  =  m  (k*  +  a2)n  ; 
whence, 

R'  =  -^ r-i— ;     Q'  =  m — fl. 

a  -t-  o  a+  o 

Again,  since  in  the  second  period  the  bar  starts  from  rest,  we  have 

R"  +  Q"  =  maw,     R"{a  +  b)  =  m(&+&)*, 

and  also  (Art.  78), 

R"  =  eR',     whence     Q"  =  eQ'y     w  =  en, 

since  Q"  and  o>  are  the  same  functions  of  R",  which  Q'  and  n  are  of  R' . 

It  is  to  be  observed  that  in  the  equations  above  the  algebraical  signs  of  the 
angular  velocities  have  not  been  taken  into  account,  and  that  the  direction  of  <a 
is  opposite  to  that  of  n. 

Finally,  if  R  and  Q  be  the  total  impulses, 

m(k2-\-a'1)n       „  ab  —  k- 

*-(i+.)*-(i+.)_LT1i->    e=(i+,)»TTTa. 

When  ab  =  k2,  the  point  A  is  the  centre  of  percussion  and  Q  =  0.     This  agrees 
with  the  result  arrived  at  in  Art.  235. 

7.  A  bar,  revolving  as  in  Ex.  6,  strikes  against  a  sphere  whose  centre  is 
moving  with  a  velocity  U  in  a  direction  perpendicular  to  the  bar ;  find  the 
magnitudes  of  the  impulses,  and  the  velocities  of  the  bar  and  sphere,  after 
collision. 

Let  M  be  the  mass  of  the  sphere,  u'  and  u  the  velocities  of  its  centre,  and 
«'  and  ca  the  angular  velocities  of  the  bar,  at  the  end  of  the  first_  and  of  the 
second  period  of  impact ;  then,  since  the  impulses  tend  to  diminish  both  the 
velocity  of  the  centre  of  inertia  and  the  angular  velocity  of  the  bar,  we  have 

R'  +  Qf  =  ma  (n  -  a/),     R'h  =  m {k2  +  a2)(n  -  »')>     R'  =  M(n'  -  V), 
where  h  =  a  +  b. 

At  the  end  of  the  first  period  of  impact  the  relative  velocity  of  the  colliding 
points  is  zero,  and  therefore,  ha  =  u'. 


Examples.  281 


Let  fl  -  w  =  zs',  then  we  have 


h  h 

and  also 

m(&  +  a2)  w  =  J/%'  - 17)  -  if/*  (W  -  Z7)  =  ifA{A(n  -  w>  -  U] , 

hence  {w(F  +  efi)  +  Mir}  W  =  J/7, (An  -  tf). 

Again, 

Eu+  Q''  =  ma{<a'-  u),  P''h  =  m{k2+a2){a}'  -  a),  Pr  =  M{ti-u'),  &niP''  =  eR' 

Hence  -we  have 

,     ,  Mm(&  +  az)(hn-U) 


Also, 


«  =  n-  (i  +  *)w  =  n-(i  +  «) 


m(&2  +  a2)  +  MK- 

Mh  {ha  -  U) 


m{k2-+az)  +  Mh~ 


22  W(F-  +  «2)(/>n-ET) 

W=  ^iT  U+{1  +  e)  "«(*»+ «*)  +  JfA2    ' 

Here,  as  in  the  former  Example,  Q  =  0  when  the  impact  takes  place  at  the 
centre  of  percussion. 

8.  Show  that  the  results  in  Ex.  6  can  he  deduced  immediately  from  those  in 
Ex.  7. 

Make  U=  0  and  M  =  oo  in  Ex.  7. 

9.  Find  at  what  point  of  its  length  the  har  should  strike  the  sphere  in  order 
that  the  impulse  of  the  hlow  should  he  a  maximum. 

If  we  put    m  (/c2  +  a2)  =  I,  we  have  to  determine  h,  so  that 

An-  u 


I+Mh? 

shall  be  a  maximum.     Hence,  to  determine  h  we  have  the  quadratic  equation 
Malr  -  IMUh  -  la  =  0. 

Bv  assuming 

U  =  rCl,     and  I  =  Mp2, 

this  equation  becomes 

A2  -  2rh  -  p~  =  0. 

We  have  then  the  following  construction  for  the  two  values  of  h.  At  0  erect 
OP  perpendicular  to  the  bar,  and  make  it  equal  to  p,  take  OG  in  the  direction 
of  G  equal  to  r,  with  C  as  centre,  and  CP  as  radius  describe  a  circle;  it  will  meet 


282     Constrained  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

the  bar  in  the  points  required.  The  value  of  h,  which  is  greater  than  r,  makes 
the  expression  for  R  a  maximum  ;  the  other  value  of  h  makes  this  expression  a 
minimum,  but  at  the  same  time  makes  R  negative.  Thus  both  values  of  h  make 
R  irrespective  of  sign  a  maximum  ;  bnt  one  impulse  is  opposite  in  direction  to 
the  other. 

• 


B      O 


If  the  sphere,  when  struck,  has  no  velocity  in  a  direction  perpendicular  to 
the  bar,  we  have  h  =  p  when  R  is  a  maximum. 

10.  Find  the  point  of  impact  in  order  that  the  impulse  on  the  fixed  axis  should 
be  a  maximum. 

If  we  put  (k2  -f-  a2)  =  K2,  we  have  to  determine  h  so  that 

{ah-K2)(hn-U) 
mK2  +  Mh2 

shall  be  a  maximum.     We  have  then  for  h  the  quadratic 


E2{MU-  maQ)         mK2 
~~2   M(aU  +  IT-ti)       ~~M~~    ' 

r'M{aU+  K2D.)  =  K2{MU  -  man), 
and  (as  in  last  Example), 


By  assuming 


the  equation  for  h  becomes 


Mp2  =  mK2, 


h2  -  2r'h  -  p2  =  0, 


and  the  two  values  of  h  are  determined  by  a  construction  similar  to  that  of  the 
last  Example. 

If  the  fixed  axis  pass  through  the  centre  of  inertia  we  have  a  =  0,  and  the 
points  for  which  Q  is  a  maximum  coincide  with  those  for  which  R  is  a  maximum. 

If  aU+  K2Q.  =  0,  one  value  of  h  is  zero,  and  the  percussion  on  the  axis  is 
a  maximum  when  the  sphere  strikes  at  the  axis. 


Free  Motion  Parallel  to  Fixed  Plane.  283 


Section  EH. — Kinetics — Free  Motion  Parallel  to  a  Fixed 

Plane. 

237.  Equations  of  Motion. — The  motion  of  a  body 
relative  to  its  centre  of  inertia  consists  at  any  instant  of  a 
rotation  round  some  axis  through  that  point.  _  Moreover,  in 
the  case  here  considered,  this  axis  must  be  at  right  angles  to 
the  fixed  plane,  and  is  fixed  in  space  if  the  centre  of  inertia 
be  regarded  as  invariable.  Now,  by  Art.  209,^  the  motion 
relative  to  the  centre  of  inertia  is  the  same  as  if  that  point 
were  fixed  in  space,  the  forces  remaining  unaltered.  Hence, 
taking  the  plane  of  yz  for  the  fixed  plane,  we  have,  to  deter- 
mine the  motion  of  the  body,  the  equations 

3»^|=sr,   3»^=Si?,    3M22=£,      (i) 

dt~  dt"  at 

where  y  and  z  are  the  coordinates  of  the  centre  of  inertia,  k 
the  radius  of  gyration  round  an  axis  through  it  at  right 
angles  to  the  fixed  plane,  and  L  the  moment  of  the  applied 
forces. 

If  the  axis  of  rotation  through  the  centre  of  inertia  be 
always  parallel  to  a  line  fixed  in  space,  it  is  plain  that  the 
last  of  these  equations  holds  good  no  matter  whether  the 
whole  motion  of  the  body  be  parallel  to  a  fixed  plane  or  not. 
In  the  latter  case  the  only  difference  will  be  that  an  addi- 
tional equation,  viz., 

will  be  required  to  determine  the  motion  of  the  centre  of 
inertia.  In  any  case,  therefore,  the  motion  of  the  body  is 
determined,  when  we  know  the  motion  of  its  centre  of  inertia, 
and  the  angular  motion  relative  to  that  point. 

238.  Connexion  of  the  Angular  Telocity  with  the 
Velocity  of  the  Centre  of  Inertia. — As  the  motion  is 
parallel  to  a  fixed  plane,  the  parallel  section  of  the  body 
passing  through  the  centre  of  inertia  must  at  each  instant  be 
rotating  round  a  point  in  its  own  plane  (Art.  219).  If  p  be  the 


284      Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

distance  from  this  point  (the  instantaneous  centre  of  rotation) 
to  the  centre  of  inertia,  s  the  path  of  the  latter,  and  w  the 

angular  velocity,  then  pw  =  — ,  as  is  obvious.     Also  w  =  — . 


Examples. 

1.  A  body  is  moving  parallel  to  a  fixed  plane  under  the  action  of  forces 
which  are  in  equilibrium  :  show  that  the  locus  of  the  instantaneous  centre  of  ro- 
tation in  the  body  is  a  circle,  having  the  centre  of  inertia  for  centre,  and  a  radius 
v 

— ,  where  v  is  the  velocity  of  the  centre  of  inertia,  and  a>  the  angular  velocity. 

2.  The  locus  of  the  instantaneous  centre  of  rotation  in  space,  under  the 
circumstances  of  Ex.  1,  is  a  straight  line  parallel  to  the  path  of  the  centre  of 

inertia,  and  at  a  distance  from  it  equal  to  -. 

CO 

3.  If  a  body  move  parallel  to  a  fixed  plane,  and  be  acted  on  by  a  constant 
couple,  lying  in  the  plane  ;  show  that  the  locus  of  the  instantaneous  centre  of 
rotation  in  space  is  an  equilateral  hyperbola. 

4.  An  inextensible  string,  whose  mass  is  negligible,  passes  over  the  line  of 
intersection  of  two  smooth  inclined  planes.  Each  end  of  the  string  passes  under 
and.  round  a  smooth  circular  homogeneous  cylinder,  to  which  it  is  attached,  and 
which  rests  on  one  of  the  inclined  planes.  The  line  of  intersection  of  the  inclined 
planes  is  parallel  to  the  axes  of  the  cylinders,  and  perpendicular  to  a  vertical 
plane  containing  their  centres  of  inertia  and  the  string.  Determine  the  tension 
of  the  string. 

As  in  Ex.  4,  Art.  230,  the  portion  of  the  string  wrapped  round  one  of  the 
cylinders  may  be  regarded  as  in  equilibrium  under  the  action  of  the  tensions  at 
its  extremities  and  of  the  pressure  produced  by  the  cylinder.  Hence  all  the 
forces  exerted  by  the  string  on  the  cylinder  are  equivalent  to  the  tension  T  act- 
ing at  the  point  of  contact  of  the  cylinder  with  the  inclined  plane. 

If  s  and  s'  be  the  distances  at  any  time  of  the  points  of  contact  of  the  cylin- 
ders and  inclined  planes,  from  the  point  of  intersection  of  the  latter  with  the 
vertical  plane  perpendicular  to  them ;  0  and  0'  the  angles  through  which  the 
cylinders  have  turned  from  their  initial  positions  ;  a  and  a'  their  radii ;  m  and  m 
their  masses  ;  and  i  and  %  the  inclinations  of  the  inclined  planes  to  the  horizon, 
the  equations  of  motion  are 


Ta, 

Ta'. 
If  a  be  the  distance  the  string  has  slipped  at  any  time  along  the  inclined 


m  ■—  =  mg  sm  %  - 

-T, 

«2    d°~d 

m  —  — 

2     dt- 

,d*s'        ,     .     ., 
tn  -j-j-  =  mgsmi 

-  T, 

,  a"2-  dH' 

m rrr 

2     dt- 

Vis  Viva,  285 

planes,  and  b  and  b'  the  initial  values  of  s  and  *',  we  have,  since  the  string  is 
inextensible, 

s  =  b  +  aQ  +  <r,     s'=  b'  +  a' 6'  —  o-,    and  therefore    s  +  s'  =  b  +  b'  +  #0  +  «'0' . 

Differentiating  twice  \vc  obtain,  by  means  of  the  equations  of  motion, 

m      .     mm'       .  .  .    ... 

T  =  I g  (sin  i  +  sin  i'). 

The  motion  can  then  be  completely  determined. 

239.  Vis  Viva. — It  was  shown  in  Art.  134,  that  the  vis 
viva  of  any  system  %mvr  =  ^flV2  +  ^mv2,  where  Tl  is  the 
entire  mass  of  the  system,  F"the  velocity  of  its  centre  of  inertia, 
and  v  the  velocity,  relative  to  the  centre  of  inertia,  of  any 
particle  m.  If  the  body  be  moving  parallel  to  a  fixed  plane, 
the  motion  relative  to  the  centre  of  inertia  is  a  rotation  round 
an  axis  fixed  in  the  body,  whose  direction  is  fixed  in  space. 
Hence  Smp'2  =  Wlk2u)2  (Art.  133),  and  the  equation  of  vis  viva 
becomes 

Wl  ( V2  +  &<f)  =  22  j  ( Ydy  +  Zdz)  +  C.  (2) 

The  equation  of  vis  viva  may  be  put  into  another  shape 
which  is  sometimes  useful.  If  I  be  the  moment  of  inertia  of 
the  body  round  the  instantaneous  axis  of  the  rotation  by 
which  the  whole  motion  of  the  body  may  be  represented,  theu 

%nv2  =  Iu>\ 

Again,  if  y  and  z  be  the  coordinates  of  any  point  referred 
to  that  space  point  as  origin  which  coincides  with  the  instan- 
taneous centre  of  rotation,  Art.  238,  then 

dyf  ,      dz' 

hence  the  equation  of  vis  viva  assumes  the  form 

therefore  *  £(/„,»)  =  j,  (3) 

Zu)  at 

where  J  is  the  moment  of  the  applied  forces,  round  the  in- 
stantaneous axis  of  rotation. 


286      Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

240.  Moment  of  the  Forces  of  Inertia. — If  b  and  c 

be  the  coordinates  of  any  point,  fixed  or  movable,  the  moment 
of  the  applied  forces,  round  an  axis  through  it  parallel  to  the 
axis  of  x,  must  be  equal  and  opposite  to  the  moment  of  the 
forces  of  inertia  round  the  same  ;  hence,  calling  the  former 
moment  J,  we  have 

aw|&r_j)__(,.0)^|.j: 

If,  as  in  Art.  209,  we  put  y  =  y  +  tj,  z  =  z  +  Z,  we  get,  by 
omitting  the  terms  which  vanish, 

^  ( ,       _ .  d"z      . _       ,  d~y     10d(u)       ,  fAS 

**{<*-*)&-{— )£+»0\->r-         (*) 

If  we  suppose  the  point  b,  c  to  coincide  with  the  origin 
fixed  in  space,  and  to  lie  in  the  plane  of  the  motion  of  the 
centre  of  inertia,  this  equation  becomes,  if  we  call  r  and  x  ^ne 
polar  coordinates  of  the  centre  of  inertia, 

241.  Moments    of    Momentum     relative     to     any 

Point. — Since  the  body  is  supposed  to  be  moving  parallel 
to  a  fixed  plane,  its  motion  at  any  instant  is  a  pure  rotation. 
If  we  take  a  line  coinciding  with  the  instantaneous  axis  of 
rotation  as  axis  of  x,  then  x,  y,  z  being  the  coordinates  of 
the  centre  of  inertia,  we  have,  by  Art.  222, 

x  =  0,     y  =  -  iw,     i  =  yta. 

Substituting  these  values  in  (31),  Art.  210,  and  introducing 
the  values  of  Hi,  R2,  E3i  given  by  (5),  Art.  231,  we  obtain 

H\={I-m{by  +  cz)}u>,      \ 

H'2=  [Way-  ^mxy}w,  [  •  (6) 

R'*  =  \Wloz  -  2mxz}u>  J 


Equations  of  Motion  for  Impulses.  287 


Examples. 

1.  The  motion  of  a  body  consists  of  a  pure  rotation ;  find  the  conditions  that 
it  should  be  brought  to  rest  by  a  single  impulse. 

Take  the  axis  of  rotation  as  the  axis  of  x,  and  a  perpendicular  p  on  it  from 
the  centre  of  inertia  G  as  that  of  y,  then  the  whole  velocity  of  G  is  parallel  to 
the  axis  of  z,  and  is  equal  to^cu,  where  co  is  the  angular  velocity  of  the  body. 
Hence  the  impulse  which  reduces  the  body  to  rest  must  be  parallel  to  the  axis 
of  z,  and  is  given  by  the  equation 

Z=  -  2Jty«. 

Let  b,  c  be  the  coordinates  of  the  point  in  which  the  impulse  Z  meets  the 
plane  of  xy ;  the  moments  of  momentum  relative  to  be  are  each  zero  after  the 
body  is  reduced  to  rest ;  but,  since  the  impulse  passes  through  be,  these  moments 
are  the  same  as  they  were  before  the  action  of  the  impulse.     Hence,  originally 

H{  =  HZ'  =  H,'  =  0. 

Substituting  for  H\,  &c,  their  values  from  (6),  we  have,  if  jBTbe  the  radius  of 
gyration  round  the  axis  of  rotation, 

K"  -  bp  =  0,     Xmxy  =  0,     2mxz  =  0. 

Hence  we  conclude  that  the  axis  of  rotation  must  be  a  principal  axis  at  the 
point  in  which  it  is  met  by  the  perpendicular  from  G,  that  the  impulse  must  be 
perpendicular  to  the  plan  containing  G  and  the  axis  of  rotation,  and  that  its 
shortest  distance  b  from  the  axis  is  given  by  the  equation 

bp  =  K2.  (Compare  Art.  235.) 

2.  A  uniform  circular  plate  whose  centre  is  fixed  lies  on  a  smooth  horizontal 
plane.  An  insect  starts  from  the  centre  of  the  plate,  and  returns  to  the  same 
point  after  describing  a  circle  whose  diameter  is  the  radius  of  the  plate ;  find 
the  angle  through  which  the  plate  has  turned. 

Let  cp  be  the  angle  through  which  the  plate  has  turned  at  any  time,  a  its 
radius,  m  its  mass,  m'  that  of  the  insect,  r  and  0  its  polar  coordinates  in  space, 
r  and  \p  its  polar  coordinates  relative  to  the  plate ;  then 

a2    dd>  „  dd 

1,1  ~   Tt+  m  '"dt^0,     ^  =  e  ~  *'     V  =  a  C0S  ^* 

__  f       2m'  cos2 \L 

Hence 


- 


on  +  2m'  cos2^/ 
and  the  angle  required  is 


_/_-_\*J. 

\m  +  2m)     ) 


242.  Equations  of  Motion  for    Impulses. — In   the 

case  of  impulses  the  changes  of  velocity  which  they  produce 
are  determined  by  the  equations  (Arts.  204,  209,  229), 

m  [v  -  t>')  =  SF,     m  {w  -  uf)  =  2Z,    Wr  {w  -  a/)  =  Z,  (7) 

where  a/  is  the  angular  velocity  of  the  body,  v  and  w  the 


288      Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

components  of  the  velocity  of  its  centre  of  inertia,  before  the 
action  of  the  impulses  ;  and  w,  v,  and  w  the  corresponding 
quantities  after  their  action. 

243.  Impact. — When  impact  occurs  between  two  smooth 
bodies,  a  mutual  impulsive  force  is  developed  in  the  direction 
of  the  common  normal.  In  the  first  period  of  collision  this 
force  reduces  the  relative  normal  velocity  of  the  colliding 
points  to  zero.  In  the  case  of  motion  parallel  to  a  fixed 
plane,  there  are  for  two  bodies  seven  unknown  quantities, 
viz.  the  changes  in  the  two  components  of  the  velocity  of  the 
centre  of  inertia,  and  in  the  velocity  of  rotation  for  each  body, 
and  the  magnitude  of  the  mutual  impulse.  There  are  like- 
wise seven  equations  to  determine  these  quantities,  viz.  the 
six  equations  of  motion,  and  the  equation  which  expresses 
that  the  relative  normal  velocity  of  the  colliding  points  is 
zero  at  the  instant  of  greatest  compression. 

In  the  second  period,  a  new  mutual  impulsive  force  is 
developed,  whose  impulse  bears  a  constant  ratio  to  that  of  the 
former,  and  can  therefore  be  found.  The  changes  of  velocity 
which  it  produces  can  then  be  determined. 

If  the  bodies  which  collide  be  perfectly  elastic,  the  im- 
pulse developed  during  the  period  of  restitution,  or  second 
period,  is  equal  to  that  developed  during  the  period  of  com- 
pression. What  is  here  stated  is  merely  a  generalization  of 
the  theory  given  in  Articles  78  and  202. 

Examples. 

1 .  A  bar,  which  is  rotating  round  an  axis  perpendicular  to  its  length,  and 
whose  centre  of  inertia  is  moving  in  a  plane  at  right  angles  to  the  axis  of  rota- 
tion, strikes  perpendicularly  against  a  fixed  obstacle ;  determine  the  impulse 
of  the  blow,  and  the  subsequent  motion. 

Let  m  be  the  mass  of  the  bar,  k  its  radius  of  gyration,  V  the  velocity  of  its 
centre  of  inertia  G  in  a  direction  perpendicular  to  its  length,  and  n  its  angular 
velocity  before  impact ;  also  let  v  and  «',  v  and  a>  be  the  corresponding  velocities 
at  the  end  of  the  first  and  of  the  second  period  of  impact,  respectively  ;  and  let 
h  be  the  distance  from  G  of  the  point  A  at  which  the  bar  strikes  the  obstacle  ; 
then,  if  R'  be  the  impulse  of  the  blow  during  the  first  period  of  impact,  and  if 
we  suppose  the  velocity  of  A  due  to  the  motion  of  translation  to  be  in  the  same 
direction  as  that  due  to  the  rotation  round  an  axis  through  C7,  we  have,  since 
the  blow  diminishes  both  the  velocity  of  translation  and  the  angular  velocity 
of  the  bar, 

JRf  =  m{V-  v'),    R'h  =  m&  (a  -  «') ; 


Examples.  289 

but  also,  since  at  the  instant  of  greatest  compression  A  is  reduced  to  rest, 
v'  +  ha  =  0.     Hence  we  obtain 

(A2  +  #)«'  =  #0-  h V\ 

.     .  ,    A(r+/*n)     ,         _,    m#(F+*n) 

therefore  n  -  co  =  _  — ,  whence  R  =  — — — — -. 

kl  +  h-  k2  +  h2 

Now,  as  in  Ex.  5,  Art,  236,  R={l  +  e)  R',  and  fl  -  «  =  (1  +  e)  (fl  -  «'), 
^_ v  =  (1  +  e>)  ( V -  v).     Hence  we  have  (k2  +  h2) R  =  (1  4«)«A2(r+  An)  ; 

(h2  -  ek2)  V-{1  +e)k2  ha              {&  -  eh2)  a-(l  +e)  hF 
consequently    v  =  —  ^  +  ^  -,     «  =  -       — ^-^ . 

2.  Find  the  point  at  which  the  bar  in  Ex.  1  should  strike  the  obstacle  in 
order  that  the  impulse  of  the  blow  should  be  a  maximum. 

We  have  here  to  determine  h  so  that  — — —  shall  be  a  maximum,  and  the 

k--\-h-  y 

required  values  of  h  are  given  by  the  quadratic  equation  h2  -f  2  —  h  —  k2  =  0, 

or  h2  +  2rh  -  k2  =  0,  if  we  put  r&  =  V.  If  C  be  the  instantaneous  centre  of 
rotation  of  the  bar,  corresponding  to  V  and  fl,  we  have  GC  =  —  r,  and  the 
points  of  the  bar  at  which  the  impact  produces  the  maximum  impulse  are 
determined  by  erecting  a  perpendicular  GP  equal  to  k,  and  with  C  as  centre, 
and  CP  as  "radius  describing  a  circle.  The  points  A  and  B  in  which  this  circle 
meets  the  bar  are  the  points  required.     (See  Fig.  p.  282.) 

Let  R\  and  R2  be  the  values  of  the  impulse  R,  corresponding  to  the  points 
A  and  JB,  respectively,  we  readily  find  that 

.Ei  =  -J^  ma  .  BG,  and  R2  =  -  l-^-  ma  .  GA. 
l  2 

The  negative  sign  of  R%  shows  that  in  this  case  the  impulse  must  act  at  the 
opposite  side  of  the  bar ;  hence,  if  we  consider  magnitude  only,  without  regard 
to  sign,  each  impulse  may  be  regarded  as  a  maximum. 

3.  A  bar  moving  as  in  Ex.  1  strikes  against  a  sphere  of  mass  M,  whose 
centre  has  a  velocity  Uin  a  direction  perpendicular  to  the  bar  ;  find  the  impulse 
of  the  blow,  and  the  subsequent  motion. 

Let  u'  and  u  be  the  velocities  of  the  centre  of  the  sphere  at  the  end  of  the 
first  and  of  the  second  period  of  impact,  then,  the  bar  beirig  supposed  to  over- 
take the  sphere,  we  have 

R  =  m  (V-  •),  R'h  =  mk2{a  -  «'),  R'  =  M  (uf  -  U)  ;       (a) 

and  also,  v'  -f  hu'  =  u' .  (b) 

If  we  put  a  -  co'  =  zs'j  from  equations  (a)  we  have 

_      ,     A2     ,     .     _      m  k2 
7-v  .-*,.-*--*, 


290      Free  Motion  of  Rigid  body  Parallel  to  Fixed  Plane. 


whence,  substituting  in  (b)  for  v  ,  «',  and  «',  we  obtain 

zj  '=  — r^>  and  therefore  R  -  (1  +  e)  —  . 

Consequently  the  motion  after  collision  is  given  by  the  equations 

&       ,  TT       /,  ^  m   k°~ 


(1  +  ^w',    v=F-(l  +  e)-?o',     w  =  *7+(l+e)  — 


4.  Find  in  Ex.  3  at  what  point  of  its  length  the  bar  should  strike  the  sphere 
in  order  that  the  impulse  of  the  blow  should  be  a  maximum. 

The  values  of  h  which  make  R  a  maximum  are  given  by  the  quadratic  equa- 
tion 


A2  + 


2rh  -  (K±J?\  k-  =  0,  where  rfl  =  V-  U. 


The  points  of  the  bar  at  which  the  impulse  of  the  blow  is  a  maximum  may  be 
determined  by  a  construction  similar  to  that  of  Ex.  2.  In  the  present  case,  C 
is  the  point  whose  velocity  perpendicular  to  the  bar  is  equal  to  that  of  the 

,    I  /M+  m\ 
sphere.     The  perpendicular  to  be  erected  at  G  is  now  kj  I — — — 1 . 

5.  In  Ex.  1  find  the  loss  of  kinetic  energy  due  to  the  impact. 

If  7'  be  the  kinetic  energy  lost  during  the  first  period  of  impact,  we  have,  by 
Ex    2   Art.  202,  27'  =  R'(V-\-  h£i),  but  if  7  be  the  total  loss  of  kinetic  energy, 

„                     ,          ox  mk°-(V+ha)2 
7  =  {l-  e*)7'  (see  Ex.  4,  Art.  202).     Hence  27  =  (1  -  e2)  — r^i • 

6.  Find  at  what  point  the  bar  should  strike  the  obstacle  in  order  that  the 
loss  of  kinetic  energy  should  be  a  maximum.  ^        ^  _  &b 

7.  In  Ex.  3  find  the  total  loss  of  kinetic  energy  of  the  system  ;  and  determine 
at  what  point  the  bar  should  strike  the  sphere  in  order  that  this  loss  should  be  a 
maximum. 

Here,  if  7'  be  the  kinetic  energy  lost  by  the  system  during  the  first  period 
of  impact,  by  Ex.  2,  Art.  202, 

27,  =  R'(V+hn)-R'U, 

Mmk\V+h£i-Uf 
hence  27  =  (1  -  O  mk2  +  M{h2  +  k2)  • 

m  +  M   &Q 

This  expression  is  a  maximum  when  h  =  — — —  . 

8.  Find  at  what  point  the  bar  should  strike  the  sphere  in  order  that  the  gain 
of  kinetic  energy  by  tbe  sphere  should  be  a  maximum.^ 

The  required  points  are  those  at  which  R  is- a  maximum. 


Examples.  291 

9.  Find  the  loss  of  kinetic  energy  by  the  bar. 

-  If  7'  be  the  loss  of  kinetic  energy  during  the  first  period  of  impact,  we  have, 
Ex.  1,  Art.  202, 

2?'  =  X'(V+hn  +  v'  +  ha'); 

TV 

but  v'+h'w'  =  «*'  =  U+jpt  and  therefore,  since  27  =  (1  -  e2)7',  we  have 

27={i-(?)^{it,+M(v+ha+ir)} 
/-,     ««-  „(   70/    v+hn-u    \2     (F+An)2-t7M 

=  (1  -  e2)  Mmk2  I  mkz  (  — ]  + — N  f . 

10.  In  Ex.  1  find  the  conditions  that  the  whole  motion  of  the  bar  should  be 
destroyed  by  the  collision. 

Am.  k2&  =  hV,  and  e  =  0.     This  is  also  easily  seen  from  first  principles. 
See  Ex.  1,  Art.  241. 

11.  A  body  is  moving  parallel  to  a  fixed  plane,  when  a  line  AB  in  the  body 
perpendicular  to  the  plane  becomes  suddenly  fixed  ;  determine  the  subsequent 
motion. 

Let  m  be  the  mass  of  the  body,  Jits  moment  of  inertia  round  AB,  k  its  radius 
of  gyration  round  a  parallel  axis  through  its  centre  of  inertia  67,  D.  the  angular 
velocity  of  the  body,  and  V  the  velocity  of  G  just  before  the  line  AB  becomes 
fixed,  p  the  shortest  distance  between  the  line  of  motion  of  G  at  this  time  and 
AB,  and  w  the  angular  velocity  of  the  body  round  AB  just  after  this  line  is  fixed, 
then  we  have 

Iv  =  m(Vp  +  k2a). 

12.  A  plane  lamina  is  moving  in  its  own  plane  when  one  of  its  points  0 
becomes  suddenly  fixed  ;  determine  the  subsequent  motion. 

Let  us  suppose  that  the  lamina  is  constrained  to  rotate  round  a  perpendi- 
cular axis  through  0,  then,  adopting  the  same  notation  as  in  Ex.  11,  we  have, 
by  (10),  Art.  234,  since  the  axis  of  rotation  is  a  principal  axis  at  0, 

/«  =  m[Vp  +  k2n),  MQ  =  0,  JV0  =  0. 

Hence  the  actual  motion  of  the  lamina  when  O  is  fixed  is  a  rotation  round  a 
perpendicular  axis,  and  the  angular  velocity  w  is  given  by  the  first  of  the 
equations  above. 

13.  A  bar  moving  in  a  vertical  plane  impinges  upon  a  smooth  horizontal 
plane  ;    find  the  motion  immediately  after  impact. 

If  the  horizontal  and  vertical  components  of  the  velocity  of  the  centre  of 
inertia  G  of  the  bar  be  represented  by  U  and  V  immediately  before  the  impact, 
and  by  u  and  v  immediately  after,  if  n  and  w  be  the  corresponding  angular 
velocities,  a  the  distance  from  Cr'to  the  point  of  impact  of  the  bar,  and  a  the 

TJ2 


292       Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

angle  which  it  makes  with  the  horizontal  plane  at  the  instant  of  impact,  the 
values  of  v  and  a  are  obtained  by  substituting  in  the  equations  of  Ex.  1,  a  cos  a 
for  h.     Accordingly  we  have 

{a2  cos2  a  —  eh"1)  V—  (1  +  e)  k2aCi  cos  a 

u  =  U,   v  = '      , — - , 

k-  +  a~  cos2  a 


_  (k2  -  ea2  cos2  a)  n  -  (1  +  e)  a  V  cos  a 
Ic2  +  a2  cos2  a 
If  the  bar  be  homogeneous,  3&2  =  a2,  and  we  get 
(3  cos2  a  -  e)  V  —  ( 1  +  e)  aCl  cos  a 


1  +  3  cos2  a 
(1  —  3<?  cos2  a)  aCl  —  (1  +  e)  V  cos  a 
(1  +  3  cos2a)tf 


14.  In  what  direction  must  an  impulse  be  applied  to  a  sphere  in  order  that  its 
initial  motion  may  be  one  of  rotation  round  a  given  tangent  ? 

The  direction  of  the  initial  motion  of  the  centre  of  inertia  of  the  sphere  is  in 
this  case  given.  Hence  the  direction  of  the  impulse  is  a  line  parallel  to  this, 
lying  in  the  plane,  which  passes  through  the  centre,  at  right  angles  to  the  given 
tangent,  and  distant  from  the  centre  by  f  radius. 

15.  A  beam  placed  in  a  smooth  horizontal  plane  is  turning  with  a  given  velo- 
city w  round  a  pivot  which  passes  through  a  given  point.  The  pivot  breaks  ; 
determine  the  subsequent  motion. 

If  b  be  the  distance  of  the  centre  of  inertia  of  the  beam  from  the  pivot,  this 
point  of  the  beam  continues  to  move  with  a  constant  velocity  bw  in  the  straight 
line  which  is  at  right  angles  to  the  beam  at  the  moment  when  the  pivot  breaks, 
and  the  beam  rotates  with  a  constant  angular  velocity  co  round  a  vertical  axis 
through  its  centre  of  inertia. 

16.  A  uniform  bar,  resting  on  a  smooth  horizontal  table,  revolves  round 
a  vertical  axis  through  its  middle  point.  The  bar  suddenly  snaps  at  its  middle 
point.     Determine  the  subsequent  motion  of  the  parts. 

17.  In  the  same  case,  find  the  point  of  its  length,  at  which  either  half  of  the 
bar  would  strike  perpendicularly  against  a  fixed  obstacle  with  the  greatest  force 
of  percussion. 

18.  Assuming  that  the  Earth's  orbit  is  circular,  show  that  its  motion,  both 
of  translation  and  of  rotation,  could  be  destroyed  .by  a  sudden  impulse  applied 
when  the  Earth  is  in  a  solstice. 

19.  Assuming  the  Earth  to  be  a  homogeneous  sphere,  calculate  in  the  pre- 
ceding Example  the  distance  from  the  Earth's  centre  of  the  line  of  action  of  the 
required  impulse.  Arts.  24  miles,  approximately. 

244.  Stress  in  Initial  Motion.  —  Stresses  are  de- 
termined, as  we  have  seen,  by  using  the  dynamical  equa- 
tions for  a  free  body,  and  introducing  unknown  reactions 
instead  of  the  geometrical  conditions.  In  many  cases  where 
the  general  equations  of  motion  cannot  be  integrated,  the 
initial  stresses  may  be  obtained  by  differentiating  the  geo- 


Examples.  293 

metrical  equations  twice,  and  introducing  into  the  equations 
thus  obtained  the  initial  values  of  the  coordinates  and  of 
their  differential  coefficients  with  respect  to  the  time,  which 
are  supposed  to  be  given.  The  initial  values  of  the  accelera- 
tions are  then  in  general  determined,  and  thence  the  un- 
known reactions,  by  means  of  the  dynamical  equations. 


Examples. 

1.  A  lamina  is  suspended  by  strings  attached  to  two  of  its  points  A  and  B, 
not  in  the  same  straight  line  -with  its  centre  of  inertia,  and  fastened  to  two  fixed 
points  0  and  0'.  The  string  joining  0'  to  B  is  cut ;  determine  the  initial  tension 
of  the  other. 


The  plane  of  the  lamina  in  its  position  of  equilibrium  must  pass  through  the 
points  0  and  0',  and  the  subsequent  motion  will  take  place  in  this  plane,  which 
we  shall  take  as  the  plane  of  yz,  the  axis  of  y  being  horizontal,  and  the  positive 
direction  of  z  downwards,  the  origin  being  0.  Let  G  be  the  centre  of  inertia  of 
the  lamina,  <p  and  0  the  angles  which  OA  and  AG  make  with  the  axis  of  y  at 
any  time,  I  and  a  the  lengths  of  OA  and  AG,  m  the  mass  of  the  lamina,  h  its 
radius  of  gyration  round  a  perpendicular  axis  through  G,  and  y  and  z  the  coordi- 
nates of  G  ;  then,  if  T  be  the  tension  of  the  string  OA  at  any  time,  we  have 

mk    -[p  =-aTsm(d-<p),  (a) 


in 


d2y         m  d2z 

-jw  =  ~  T  cos  4>>     m  jt  =  »*ff  -  T sin  <p.  (b) 


Also,  y  =  lcos(p  +  a  cos  6,     z  ~  Isincp  +  a  sin  6.  (e) 

Differentiating  these  latter  equations  twice,  and  in  the  second  differentiation 
treating  6  and  <p  as  constants,  since  initially  —  and  ~  are  each  zero,  and  finally 

(It  (it 

substituting  their  initial  values  a  and  £  for  6  and  <p,  we  obtain 

d~V      ,  •    „d°~<l>  ■       d~9      d2z      ,  d2<f>  d°~9 


294     Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

d2<t> 
Hence,  eliminating  -— ,  we  get 

d2z  d?y  .    .        nSd29      A 

sin  j8  — -  +  cos j8  —  +  «sin(a  -  0)  ■—  =  0. 


dt2 


df- 


dt~ 


Substituting  from  (a)  and  (b),  and  putting  a  and  £  for  0  and  (/>  in  those  equa- 
tions, we  get  for  To,  the  initial  value  of  the  tension, 


To  =  mg 


k*  sin  j8 


A-2  +  «2sin2(a-£) 


2.  A  body,  whose  centre  of  inertia  is  G,  is  suspended  by  strings  attached  to 
two  of  its  points  A  and  B,  and  fastened  to  two  fixed  points  0  and  0' .  The 
plane  AGB  is  a  principal  plane  at  G,  the  string  O'B  is  cut;  determine  the 
initial  tension  of  the  other. 

We  may  here  suppose  the  body  compelled  to  rotate  round  an  axis  through  G, 
whose  direction  is  fixed  in  space,  and  is  perpendicular  to  the  initial  position  of 
the  plane  AGB.  Since  this  axis  is  a  principal  axis  at  G,  we  find,  then,  Art.  236, 
equation  (14),  that  the  components  of  the  stress  couple  on  this  axis  are  zero,  and 
therefore  that  the  body  rotates  round  it  freely.  Hence  the  whole  motion  of  the 
body  is  parallel  to  the  vertical  plane  which  is  the  initial  position  of  AGB,  and 
the  question  becomes  the  same  as  in  the  last  example. 

3.  A  circular  disk  is  hung,  with  its  plane  horizontal,  from  a  fixed  point 
vertically  over  its  centre,  by  means  of  three  equal  strings  attached  to  three  fixed 
points  in  the  circumference  of  the  disk  at  equal  distances  from  each  other.  One 
of  the  strings  is  cut ;  determine  the  initial  tensions  of  the  other  two. 

The  two  tensions  along  the  threads  OA  and  OB  may  be  replaced  by  the 
single  force  F  along  OS,  where  F  =  2 T  cos  AOS,  S  being  the  middle  point  of 
the  chord  joining  the  fixed  points  A  aniLB. 


In  this  case  F  takes  the  place  of  T,  and  the  point  S  of  A  in  Ex.  2.  Then, 
&  being  the  initial  value  of  the  angle  which  OS  makes  with  the  horizontal  line 
which  is  the  initial  direction  of  SG,  the  length  of  the'latter  being  a,  we  have, 
since  SG  is  originally  horizontal, 

k2  sin  £ 

F=m»  e  +  oW 


Examples. 


295 


If  l  be  the  length  of  OS,  the  expression  for  F  may  be  put  into  the  form 

k~  sin  £ 


F=  mg 


A--  +  £-sin2/3  cos-)3 


4.  Determine  in  Ex.  9,  Art.  202,  the  initial  tensions  of  the  strings,  and 
their  tensions  when  the  bar  is  at  its  greatest  height,  the  length  of  each  string 
being  2a. 

If  9  be  the  angle  one  of  the  strings  makes  with  a  vertical  line  at  any  time, 
z  the  vertical  coordinate  of  the  middle  point  of  the  bar,  i//  the  angle  the  bar 
makes  with  a  horizontal  line  parallel  to  the  fixed  bar,  and  T  the  tension  of  one 
string ;  then 

or  d2ib  „    _   .  .  , 

m --?-  =  - 2a  T  sm  6  cos  ^, 

3    at- 


m  —  =  2Tcos 
dt~ 


mg  ; 


also,  from  the  geometrical  conditions, 


2i|/,  as  2asin0  =  2«sin^, 


z  =  2a(l-cos0). 
Substituting  £  \\>  for  0  in  the  last  equation,  differen- 
tiating twice,   and  observing  that  initially  \p  =  0,  and  —  =  a>,  we  get.  for  the 


d$ 


initial  tension  of  one  string  T  =  J  mg  +  f  maw 

To  get  the  tension  when  the  bar  is  at  its  highest  position,  make  -j-  =  0 

.  ,       2a  -  h 
cos  h  ^  = 

2a 

T=mg 


-,  where  h  has  the  value  in  Ex.  9,  Art.  202 ;  then 
4«3  288<73 


=  mg 


( 1 2g  -  aw2)  { 48^3+ft»2  (24ag  -  a?  w2) } 


{2a-h){ia2+Sh(4a-h)} 

5.  In  Ex.  1,  find  the  values  of  a  and  £,  in  order  that  T0  shall  be  the  greatest 
possible. 


Ans. 


£  =  -.     The  corresponding  value  of  T0  is  mg,  i.e.  the  weight  of 


the  lamina. 

6.  If  £  be  given,  find  a,  so  that  T0  shall  be  a  minimum. 

Here  sin  (a-  0)  =max.,  and  therefore  a  -  £  =  |,  or  -40  is  perpendicular 

to  CL4. 

7.  If  the  initial  position  of  AG  be  horizontal,  find  £,  so  that  To  shall  be  a 


smjS  1 

Here  Ave  have  to  find  /3,  so  that  — — — 1-  -=-: — - 

¥■         c2sm)8 


may  be  a  minimum.  There- 
fore, k  =  a  sin  £  =  p0,  where  p0  is  the  initial  value  of  the  perpendicular  from  G 
on  OA. 

The  result  here  obtained  holds  good  for  Ex.  3,  if  OS  be  substituted  for  OA, 
and  F  for  T. 


296     Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

245.  Friction. — Friction  [see  Art.  60)  is  a  tangential 
force  passing  through  the  point  of  contact  of  two  rough 
surfaces,  which  tends  to  prevent  the  one  from  slipping 
on  the  other.  If  there  be  slipping,  the  friction  is  in  an 
opposite  direction,  and  takes  its  greatest  possible  value,  which 
is  in  a  constant  ratio  to  the  normal  pressure  between  the 
surfaces.  If  the  motion  be  pure  rolling,  just  enough 
friction  is  exerted  to  maintain  pure  rolling.  The  force  of 
friction  is  then  usually  less  than  its  maximum  value,  and  is 
determined,  as  if  it  were  an  unknown  reaction,  by  means  of 
the  equations  of  motion  and  the  geometrical  condition  which 
expresses  that  the  motion  is  pure  rolling.  If  the  value  thus 
found  for  the  force  of  friction  does  not  exceed  its  maximum 
value,  and  pure  rolling  be  consistent  with  the  initial  condi- 
tions, it  will  be  the  actual  motion.  When  there  is  slipping, 
the  friction,  which  is  then  a  maximum,  and  therefore  de- 
termined, tends  to  make  the  motion  pure  rolling.  If  pure 
rolling  be  attained,  the  friction  at  the  instant  pure  rolling 
commences  changes  in  general  its  value,  and  must  be  de- 
termined in  the  manner  stated  above. 

It  is  to  be  observed,  as  already  stated  in  Art.  60,  that  the 
maximum  value  of  friction,  when  slipping  actually  takes 
place,  is,  in  general,  less  than  its  maximum  value  when  there 
is  no  slipping,  and  friction  is  acting  against  a  force  which 
tends  to  produce  slipping. 

When  a  surface  is  said  to  be  perfectly  rough  it  is  under- 
stood that  ncr  slipping  can  take  place  between  it  and  any 
other  surface  with  which  it  is  in  contact.  The  amount  of 
force  which  it  is  capable  of  exerting  by  means  of  friction  is, 
in  this  case,  unlimited. 

Examples. 

1.  A  homogeneous  cylinder,  having  its  axis  horizontal,  rolls  without  slipping 
down  a  rough  inclined  plane ;  determine  the  amount  of  friction  brought  into 
play  (see  Ex.  1,  p.  139). 

The  equations  of  motion  are 

M^  =  Mgsmi-F,     Mk^  =  aF; 
dt2         *  dt2 

the  axis  of  y  being  a  line  in  the  inclined  plane  at  right  angles  to  its  intersection 


Examples.  297 


with  the  horizon.     Also,  adO=  dy  ;  whence 

k2  sin  i 


F^Mg 


<l2+k2 


a2 
but  since  the  cylinder  is  homogeneous  we  have  k2  =  — , 

and  therefore  F  =  ^  Mg  sin  i. 

2.  If  a  sphere  he  substituted  for  a  cylinder  in  the  last  example,  determine 
the  amount  of  friction  brought  into  play.  Ans.  F  =  f  Mg  sin  i. 

3.  A  lamina  is  placed  on  a  rough  horizontal  table  in  such  a  manner  that  its 
centre  of  inertia  lies  beyond  the  edge  of  the  table,  and  that  the  line  in  which 
the  edge  meets  the  lamina  is  a  principal  axis  for  the  point  0  in  which  it  is  met 
by  the  perpendicular  from  the  centre  of  inertia  G ;  determine  the  motion  of  the 
lamina  before  it  slips,  and  its  inclination  to  the  table  when  slipping  begins. 

Since  the  force  tending  to  make  the  lamina  slip  is  at  first  zero,  the  motion 
of  the  lamina  begins  by  a  rotation  round  the  edge  AB  of  the  table,  as  a  fixed 
axis. 

Putting  M  for  the  mass  of  the  lamina,  and  otherwise  adopting  the  same 
notation  as  in  Ex.  3,  Art  236,  we  have,  since  a  =  0, 

Po  =  Mg    72+    P,  sin  9,      Q0  =  Mg  cos  6. 

k2  +  p-  k-  +  pl 

The  lamina  continues  to  rotate  round  AB  till  F0  =  fiQ0,  where  fx  is  the  coefficient 
of  friction.  The  value  of  9  when  the  lamina  begins  to  slip  is  given  therefore 
by  the  equation 

tan  9  = — -  u. 

k2  +  Zp2  r 

4.  In  Ex.  3  a  mass  m  is  placed  at  a  point  D  on  the  lamina,  in  the  perpen- 
dicular from  its  centre  of  inertia  on  the  edge  of  the  table  ;  investigate  the 
motion,  and  find  the  inclination  at  which  slipping  begins. 

Let  OB  =  A,  then,  since  the  initial  motion  is  a  rotation  round  AB  as  a  fixed 
axis,  we  have 

{M{k2  +  p2)  +  mh2 }  -—  =  (Mp  +  mh)  g  cos  9.         {a) 

Hence,  by  integration, 

(d9\2      n         Mp  +  mh  .    n  ... 

The  forces  acting  on  m,  in  addition  to  gravity,  are  the  force  of  friction  P  along 
DO,  and  the  resistance  Q,  perpendicular  to  BO,  of  the  plane.  Hence  the 
accelerations  of  f»,  along  BO,  and  perpendicular  thereto,  are 

F  Q 

g  sin  9 ,  and  g  cos  9 ; 

m  w 


298     Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

but,  since  OB  is  invariable  so  long  as  m  does  not  slide,  the  accelerations  of  in 
are  also  —  hu2,  and  h  — ,  by  (11)  and  (12),  Art.  28  ;  hence  we  have 


P=  m  (g  sin  0  +  hw2),     Q  =  m  I g  cos  9  —  h  ^ ) 


dw\ 
dt) 


(Mp  +  mk)  h 
If  we  put  A  = 


M  (k2+p2)  +  mh2' 
equations  (a)  and  (b)  give 


du 
h 

hence  we  get 


h  —  =  \g  cos  0,  and  hu)2  =  1\g  sin  9 
dt 


P  =  mg  sin  0  (1  +  2a),     Q  =  mg  cos  0(1-  A). 

"We  here  observe  that  Q  becomes  negative  if  A  be  greater  than  unity  ;  accord- 
ingly in  that  case  the  mass  m  is  left  behind  by  the  lamina  from  the  very  com- 
mencement of  the  motion,  unless  we  place  it  beneath  the  lamina. 

If  A  =  1,  or  ph  =  h2  +p2,  we  have  Q  =  0  :  m  is  in  this  case  placed  at  the 
centre  of  oscillation,   and  begins  to  slip  at  the  very  commencement  of  the 

motion.     If  A  <  1,  m  begins  to  slip  when  tan  d\  =  /j.  - — ■ — . 

1  +  2A 
Next,  let  jP0  and  Q0  be  the  forces  parallel  and  perpendicular  to  OG,  exerted 
against  the  edge  of  the  table,  we  have,  (15),  Art.  236,  since  the  whole  system 
at  first  is  moving  as  a  rigid  body, 

P0  =  (Mp  +  tnh)  or  +  (31  -f  m)  g  sin  0, 

Q0  =  ( M  +  m)  g  cos  0  -  (Mp  +  mh)  —. 

dt 

Mp  +  mk 

Hence,  it  v  =  -— — ,  we  readily  get 

(M  +  m)  h  y  b 

P0  =  (M-\-  m)g  sin  0(1  +  2\v),     Q0  =  (M+m)g  cos  0(1  -Ac). 

If  the  coefficient  of  friction  relative  to  the  lamina  be  the  same  for  the  edge 
as  it  is  for  m,  the  lamina  begins  to  slip  when  P0  =  /j.Q0,  or  when 

tan  0O  =  ^r ■• 

1  +  2aj/ 

Hence,  if  v  <  1,  i.e.  if  h  > p,  we  have  0o  >  0i,  and  therefore  in  this  case  the 
mass  will  slip  before  the  lamina  begins  to  slip. 

On  the  other  hand,  if  h  <  p,  we  have  0i  >  0Oa  and  slipping  begins  at  the 
edge  AB. 


Examples.  299 

5.  In  Ex.  3,  if  any  number  of  masses  mi,  m2,  &c,  be  placed  on  tbe  lamina 
at  points  Du  D2,  &c,  on  the  line  OG,  investigate  the  motion. 
Let     ODi  =  hh  01*2=}%  &c,  then, 

d(o  Mp  +  mi  7*i  +  m2h2  +  Sec. 

= ; — — -  g  cos  0, 

dt       M(k2  +p2)  +  mi  hr  -t-  m2  hi~  +  &c. 

Mp  +  tnihi  +  m2h2  +  &c.  . 

W=     Jf  (&2  +i?2)  +  mi  fa2  +  m2 h22  +  &c.9  Sm 

If  we  put 

Mp  +  mihi  +  m2?i2  +  Szc.         _  Ai  _  \2  _ 

M(k2  +p2)  +  mi  7*r  +  mzfa2  +  &c.  ~  h  ~  h2  ~       '' 


we  have 


and  also 


Mp  +  mihi  +  rnhz  +  Scc.  . 

,, c =  vihi  =  v2  h2  =  &c. , 

M  +  mi  +  w?2  +  &c. 

Pi  =  (1  +  2Ai)  mi^  sin  0,     Qi  =  (1  —  Ai)  miy  cos  9  ; 

JP2  =  (1  +  2\2)m2g  sin  0,      Q%  =  (1  -  A2)  m2<7  cos  0  ; 
&c,  &c.  ; 

p 

1  +  2Aij/i  =  1  +  2A2n=  &c. 


(M  +  mi  +  m2  +  &c.) g  sin  0 


,      - —  =  1  —     \\v\  =  1  —     A2J/2  =  &C 

(M  +  mi  +  m2  +  &c.)  g  cos  0 

The  rest  of  the  investigation  is  the  same  as  in  the  last  example. 

_     1  -  Ai          1  -  A2 
If      hi  >  h2,  then  Ai  >  A2,  and  <- — -, 

1  -f-  Z\i         1  +  ZA2 

and  therefore  9\  <  62,  or  mi  slips  before  m2  :  that  is,  the  mass  farthest  from  the 
edge  begins  to  slip  first. 

6.  If  a  hoop  rolls  down  a  rough  inclined  plane  without  sliding,  show  that 
tan  i  <  2/jl  ;  the  initial  position  of  the  hoop  being  in  a  vertical  plane  at  right 
angles  to  the  intersection  of  tbe  inclined  plane  with  the  horizon. 

Take  the  initial  position  of  the  centre  of  the  hoop  for  origin,  and  the  inter- 
section of  the  inclined  plane  with  a  vertical  plane  at  right  angles  thereto  as  axis 
of  y,  its  positive  direction  being  downwards.  Let  the  positive  direction  of  rota- 
tion be  from  the  upper  side  of  the  inclined  plane  towards  y  positive.  Then, 
y  being  the  coordinate  of  the  centre  of  the  hoop,  m  its  mass,  a  its  radius,  and 
Pthe  friction  brought  into  play,  the  equations  of  motion  are 

n  d&  dry  .     .       -, 

ma2  —  =  Fa,     m  — r  =  mg  sm  i  -  F; 
at  at- 


300     Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

but  the  motion  being  pure  rolling,  aw  =  —  ;  hence,  eliminating  we  obtain, 


F  = 


sini 


but  F<  fiing  cos  i ;  therefore  tan  i  >  2fi. 


7.  A  homogeneous  circular  disk,  whose  radius  is  a,  rolls  inside  a  rough  ver- 
tical circle  whose  radius  is  b  ;  the  motion  is 
pure  rolling  under  the  action  of  gravity ; 
show  that  the  rolling  forward  and  backward 
of  the  disk  is  isochronous  with  the  oscilla- 
tions of  a  simple  pendulum  whose  length 
is  f  {b  -  a). 

¥e  have,  I  being  the  moment  of  inertia 
of  the  disk  round  an  axis  through  P,  w  the 
angular  velocity,  and  9  the  angle  between 

GA  and  GP,  -  -  {lor)  =  2mga  sin  9  (0  being 
co  at 

reckoned  from  the  vertical  line  GA,  where  G 

is  the  centre  of  the  vertical  circle,  and  w  being  the  angular  velocity  of  the  disk 

rolling  down).     As  the  instantaneous  centre  of  rotation  lies,  in  this  case,  on  the 

circumference  of  the  disk,  I  remains  constant  throughout  the  motion ;  there- 

jlw 


fore  I 


dt 


mga  sin 


but   I  = 


(Integral   Calculus,  Chap.  X.),  and 


aw  =  —  (b  —  a)  — ,  since  either  represents  the  velocity  of  0.     Hence 
dt 

d~9 
§  (b  —  a)  —  =  -  g  sin  9  ;   .'.  &c. 

The  student  will  observe  that  the  friction  at  P  does  not  enter  this  equation. 

8.  A  uniform  sphere,  resting  on  a  rough  horizontal  plane,  is  set  in  motion  by 
an  impulse  applied  in  a  vertical  plane  passing  through  its  centre.  Show  that, 
when  sliding  ceases,  the  rolling  motion  will  be  direct,  stationary,  or  retrograde, 
according  as  the  direction  of  the  impulse  intersects  the  vertical  diameter  above, 
at,  or  below  the  point  of  contact  with  the  plane. 

Let  v  be  the  velocity,  at  any  time,  of  the  centre  of  the  sphere  parallel  to  the 
intersection  of  the  horizontal  plane  with  the  vertical  plane  containing  the 
impulse ;  the  direction  of  the  latter  making  an  acute  angle  with  the  positive 
direction  of  v.  Let  w  be  the  angular  velocity  of  the  sphere,  counted  from  the 
vertical  towards  the  direction  of  v  positive  :  then  V  and  n,  the  initial  values  of 
v  and  w,  are  determined  by  the  equations 

mV  =  Y,     ink-  n  =  Yb, 


where  Y  is  the  horizontal  component  of  the  impulse,  and  b  the  distance  from 
the  centre,  at  which  its  line  of  direction  intersects  the  vertical  diameter  of  the 
sphere.     Eliminating  Y,  we  obtain 


a  = 


b_V 
£2' 


Examples.  301 

For  the  subsequent  motion,  if  Fbe  the  force  of  friction,  we  have  the  equations 

dv      -„  *o  do  _ 

m  —  =  F,      mk~  —  =  -  Fa  ; 
dt  dt 

dv        7.  dw 
whence  a  —  +  r  — =0. 

Integrating,  we  obtain 

av  +  k2co=  constant  =  a  V  +  £2n  =  (a  +  b)  V. 

"When  sliding  ceases  v  =  aw.  Substituting  for  v  in  the  preceding  equation,  we 
have 

(a+  b)V 

<0  =  5 rr-. 

Hence,  since  Fis  necessarily  positive,  w,  when  sliding  ceases,  is  positive,  zero, 
or  negative,  according  as 

a  +  b>  0,     «  4  J  =  0,     or     a  +  b  <  0. 

The  first  condition  holds  good,  if  b  is  either  positive,  or  negative  and  less  than  a 
in  absolute  magnitude  ;  the  second,  if  b  =  —  a  ;  the  third,  if  b  is  negative  and 
greater  than  a  in  absolute  magnitude. 

The  results  of  this  example  may  be  extended  to  other  solids  of  revolution. 

9.  A  circular  plate  rolls  down  the  inner  circumference  of  a  rough  circle  under 
the  action  of  gravity.  The  plane  of  the  plate  coincides  with  that  of  the  rough 
circle,  which  is  vertical.  Determine  the  amount  of  friction  brought  into  play 
if  the  plate  start  from  rest,  the  motion  being  pure  rolling.     (See  Ex.  7.) 

If  co  be  the  angular  velocity  of  the  plate,  the  equations  of  motion  are 

da  d2Q  .  ,., 

\ma2  —  =  Fa,      m  ih"a)j^  =  -  m9  sm  d  +  F 

together  with  the  equation  of  condition 

dQ 

hence  F=  \mg  sin  9. 

10.  Show  that  the  plate  in  the  last  example  will  ascend  to  the  same  height 
as  that  from  which  it  started,  and  that  the  motion  will  go  on  for  ever. 

The  vis  viva  =  2mg  (z  -  z0)  :  this  will  vanish  when  z  =  z0 ;  therefore,  &c. 

11.  Determine  the  velocity  of  rotation  of  the  plate  at  any  time. 

Ans.    u)2  =  i      *"„     ■  (cos  6  -  cos  0O). 


v- 


302      Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

246.  Tendency  of  a  JRod  to  Break. — When  a  body 
is  under  the  influence  of  any  forces,  it  experiences  pressures 
or  tensions,  which  tend  to  alter  the  relative  positions  of  the 
molecules.  This  tendency  is  resisted  by  the  mutual  action 
of  the  molecules.  Under  such  circumstances  the  body  is 
said  to  be  in  a  state  of  stress. 

If  we  consider  a  small  rectangular  parallelepiped  in  the 
body,  the  stresses  acting  on  one  of  its  faces  may  be  resolved 
into  three  forces  at  right  angles  to  each  other — one  normal, 
and  two  parallel  to  the  face  under  consideration. 

To  ascertain  the  tendency  of  a  body  to  undergo  a  rupture 
in  any  part,  we  must  consider  the  stresses  to  which  it  is  sub- 
jected in  that  part.  If  the  mutual  cohesion  of  the  molecules 
is  unable  to  resist  these  stresses  the  body  must  give  way. 
The  question  is,  in  general,  one  of  great  complication,  and 
for  its  full  discussion  the  reader  is  referred  to  treatises  on 
Elasticity  and  Strength  of  Materials. 

If  the  body  under  consideration  be  a  rod,  that  is,  if  two  of 
its  dimensions  are  at  each  point  very  small,  the  question  be- 
comes much  simplified. 

The  axis  of  the  rod  may  be  a  straight  line,  or  may  form 
a  curve  of  any  kind.  We  shall  suppose  that  this  curve  is  not 
closed,  that  it  lies  in  one  plane  P,  and  that  the  rod  is  in 
equilibrium  under  the  action  of  forces  in  this  plane.  If  we 
consider  a  section  at  right  angles  to  the  axis  of  the  rod,  at 
any  point  A  of  its  length,  the  action  of  the  molecules  at  one 
side  of  this  section  on  those  at  the  other  must  equilibrate  all 
the  forces  acting  on  the  rod  at  the  latter  side.  These  may  be 
reduced  to  a  force  F,  passing  through  A,  and  a  couple  G, 
round  an  axis  a  at  right  angles  to  the  plane  P.  This  force 
and  couple,  therefore,  are  equivalent  to  the  stresses  acting  on 
the  rod  through  the  section  containing  A. 

That  the  tendency  of  the  rod  to  break  results  chiefly  from 
the  couple  may  be  shown  as  follows : — 

The  stresses  in  the  plane  of  the  section  cannot  give  any 
couple  round  the  axis  a,  since  a  either  meets  them  or  is  parallel 
to  them.  Hence  the  couple  G  must  produce  stresses,  parallel 
to  the  axis  of  the  rod  at  the  point  A,  whose  moment  round  A 
is  equal  to  G.     If  iV  be  the  value  per  unit  of  area  of  the 


Tendency  of  a  Rod  to  Break.  303 

greatest  of  these  stresses,  and  a  be  the  distance  from  A  of  the 

most  remote  point  of  the  section,  whose  area  may  be  denoted 

by  8;  the  moment  round  A  of  the  stresses  parallel  to  the 

axis  must  be  less  than  NSa.     Hence,  if  we  assume  G  =  Fp, 

we  have 

F  p 
NSa  >  Fp,  and  therefore  N  >  -=■-. 

o  a 

If  we  now  seek  for  the  stress  per  unit  of  area  caused  by 

XT'  n 

the  force  F,  we  have  N'  =  -;    .:  N'  <  -  N. 

S  p 

Hence,  if  a  is  very  small  compared  with  p,  J¥'  is  unim- 
portant compared  with  iV.  Accordingly,  in  general,  the 
tendency  of  the  rod  to  break  at  any  point  A  depends  simply 
on  N,  i.e.  on  G,  the  moment  round  A  of  the  forces  acting  on 
the  rod  at  one  side  of  A. 

We  have  hitherto  supposed  the  rod  to  be  at  rest.  If  it 
be  in  motion,  we  can,  by  D'Alembert's  Principle,  consider  it 
as  in  equilibrium  under  the  action  of  the  applied  forces  and  the 
forces  of  inertia,  and  the  question  of  stress,  or  the  tendency  to 
break  at  any  point,  becomes  the  same  as  before,  except  that 
we  must  now  add  the  forces  of  inertia  to  the  other  forces 
acting  on  the  rod. 

If  the  rod  be  acted  on  by  impulses,  the  impulsive  tendency 
to  break  at  any  point  is  obtained  in  a  similar  manner,  and 
the  preceding  investigation  holds  good  provided  the  impulses 
be  substituted  for  the  applied  forces,  and  the  resulting 
changes  of  momentum  for  the  forces  of  inertia. 

To  find  the  couple  which  measures  the  tendency  of  a  rod  to 
break  at  any  point  P. 

Let  G  be  the  required  couple,  L'  the  moment  round  P  of 
the  forces  applied  to  the  portion  of  the  rod  on  one  side  of  this 
point,  tyl'  the  mass  of  this  portion  of  the  rod,  kf  its  radius  of 
gyration  round  its  own  centre  of  inertia  C,  and  A'  the  moment 
of  the  acceleration  of  C  round  P,  theu  by  (4) ,  Art.  240, 


e -J? -*(*'♦*•$}  (8) 


304     Free  Motion  of  Rigid  Body  Parallel  to  Fixed  Plane. 

In  the  case  of  impulses,  if  G  be  the  impulsive  couple  cor- 
responding to  G,  we  have 

G  =  L'-m'{A'+k'>  (w-o/)),  (9) 

where  A'  is  the  moment  round  P  of  the  change  of  velocity  of 
C  due  to  the  impulses,  and  w  and  a/  are  the  angular  veloci- 
ties of  the  rod  after  and  before  the  action  of  the  impulses. 

Another  expression  for  G  which  is  often  useful  may  be 
found  as  follows  : — Let  //,  z  be  the  coordinates,  referred  to  a 
fixed  origin,  of  the  centre  of  inertia  C  of  the  whole  rod ;  b,  c 
those  of  P ;  y,  z  those  of  C\  and  r,,  Z  those  of  any  point  of 
the  rod  referred  to  axes  through  C  parallel  to  the  fixed  axes, 
then, 

G=  L'-Z'm{(y-b)z-  {z-c)ij}. 

But  &  =  $+'*,    z  =  z  +  Z; 

substituting,  and  remembering  that 

^!my  =  9W//,     Sm  =  Wl'  z\ 
we  obtain 

G  =  L'-W{(y-b)'i-(z'-c)$}-2'm{(y-b)Z-(z-cyri}.  (10) 

In  the  case  of  impulses  applied  to  a  rod  at  rest, 
G  =  2/-3RV-  6)5- (s' -  e)  f}-Xm{{y-b)  £-  («-  *)$}.  (11) 

If  the  rod  be  in  motion  when  the  impulses  are  applied,  we 
must  substitute  in  (11)  for  f,  £,  17,  and  £  the  changes  in  their 
values  due  to  the  action  of  the  impulses. 

Examples. 

1.  A  uniform  straight  rod  AB  rotating  round  a  perpendicular  axis  passing 
through  one  extremity  A  is  struck  perpendicularly  at  a  point  Q  ;  find  the  ten- 
dency to  break  at  any  point  P. 

Let  H  be  the  impulse  of  the  blow,  a  the  length  of  the  rod,  m  its  mass,  «'  and 
&>  its  angular  velocities  before  and  after  the  blow  ;  also  let  C  be  the  middle  point 
of  PB,  and  let  AP=  r,  AQ  =  h  ;  then, 

L'  =  P.PQ,     A'  =  PC'.AC  {co-o}'),     M'*  =  PC'2,     m'a  =  2mPC,) 


Examples.  305 

hence  we  have 

n.  -  7?    vn 

da 


2«i 

G  =  E.PQ-'—  PC'HZAC  +  PC)(co  -  «') 


But  m— {(o -u')  =  H.  AQ  =  Rh, 

u 

a  —  r         .  „,     a  +  r 
PQ  =  h-r,     PC  =  -^-i     AC=^-. 

Substituting  these  values  in  the  equation  for  G  we  obtain 

G  =  £r  {2a*(h  -  r)  -  k(a  -  r)»  (2a  +  r)} 
2  a6 

2  a6 

2.  In  Ex.  1  find  the  position  of  the  point  at  which  the  tendency  to  break  is 
a  maximum. 

If  3A  >  2a,  the  tendency  to  break  is  a  maximum  when 

3.  A  uniform  rod  is  turning  in  a  vertical  plane  round  a  horizontal  pivot  A, 
at  one  of  its  extremities.     Find  the  tendency  to  break  at  any  point  P. 

Adopting  the  same  notation  as  in  Ex.  1,  and  denoting  by  0  the  angle  which 
the  rod  makes  with  the  horizontal  line,  we  have 

a  ~  r     , 
L  =  — - —  m  g  cos  9. 
z 

Moreover,  since  C  is  moving  in  a  circle  round  A  as  centre,  its  acceleration  has 
two  components — one  at  right  angles  to  PC,  which  is 

a+  r  d°-d 


2      W 

and  the  other  along  PC.     The  latter  gives  no  moment  round  P ;  hence 

a  +  r  a  -  r  d29 

A  =  ~~2     r~  w 

and  G^-j-mgcme-m   |  __—+*»—[; 

but  |  w«2  —  =  £»#«  cos  0,  and  A-  -  =  — ^—  ; 

(a  -  rY- 
whence  G  =  -mg  ■    4q2     ♦*  cos  0. 

X 


306     Free  Motion  of  Rigid  Body  parallel  to  Fixed  Plane. 

4.  A  cracked  hoop  rolls  on  a  perfectly  rough  horizontal  plane.  Determine 
the  inclination  to  the  horizon  of  the  line  joining  the  crack  to  the  opposite  point 
when  the  tendency  to  hreak  at  this  point  is  the  greatest  possible. 


In  this  case  the  centre  of  inertia  of  the  hoop  moves  in  a  straight  line  with  a 
constant  velocity.  Hence  its  acceleration  is  zero  ;  also  if  a  be  the  radius  of  the 
hoop, 

cos  9  +  —  sin  9  ) ,  since  CO  =  — , 


J=Tg\a 


O  being  the  centre  of  inertia  of  the  semi-hoop  comprised  between  the  crack  Q 
and  the  opposite  point. 

Now,  since  the  angular  velocity  round  a  horizontal  axis  through  G  is  con- 
stant, the  system  of  forces  m'TJ,  m(,  &c.  in  (10),  are  equivalent  to  a  single  force 
STar .  GO  in  the   direction  of  CO.      The  moment  of  this   force  round  P  is 

M^—  a-,  which  is  independent  of  9.     The  tendency  to  break  at'P  is  given  by 

IT 

the  equation 

^      «-  (      /cos  0      sin0\        a1  a2) 

ff.jr|<r(_  +  _)-_J. 

2 
Hence  the  tendency  to  break  is   a  maximum   when   tan  9  =  -,    provided 

g  (2  +  VV3  +  4)  >  4rtar.     This  condition  appears  by  considering  when  G  attains 
its  greatest  magnitude,  irrespective  of  sign,  if  it  should  become  negative. 

5.  In  Ex.  3  find  at  what  point  of  the  rod  the  tendency  to  break  is  a  maxi- 
mum. Ans.  r  =  ia. 

6.  A  semicircular  wire,  of  radius  a,  lying  on  a  smooth  horizontal  table,  turns 
round  one  extremity  A,  with  a  constant  angular  velocity  o>.  Find  the  tendency 
to  break  at  any  point  P. 

Let  0  be  the  centre  of  inertia  of  the  arc  PB,  and  let  PGA  =  (p.  Join  AO, 
AP,  and  PO  ;  then,  since  the  angular  velocity  is  constant,  the  acceleration  of  0 

is  co3  .  AO.     Consequently  —  is  double  the  area  of  the  triangle  APO ;  but  since 

AP  and  GO  are  parallel,  the  triangle  APO  is  equal  to  the  triangle  AGP; 

hence  A'  =  a2 or  sin  <p,     and     G  =  m  — a'-  or  sin  (p. 


Examples.  307 

Accordingly  the  tendency  to  break  is  a  maximum  at  the  point  determined  b y 
the  equation  tan  <$>  =  -n  -  <p. 


This  example,  as  well  as  3  and  4,  are  taken  from  Kouth,  Rigid  Dynamics. 

7.  A  free  uniform  straight  rod  is  set  in  motion  by  a  perpendicular  impulse  ; 
find  the  tendency  to  break  at  any  point  P. 

Let  AB  be  the  rod,  C  its  middle  point,  M  its  mass,  2a  its  length,  Q  the  point 
at  which  it  receives  the  impulse  R,  C  the  middle  point  of  PB,  M'  its  mass, 
v  the  velocity  of  C,  and  a>  the  angular  velocity  of  the  rod  after  the  action  of  the 
impulse  ;  let  CP  =  r,   CQ  =  h,  then 

L=R.PQ,     {y'-b)-z-{z'-c)y-  =  PC'.v, 
2'm  {(y-b)  C- {z-c)-h}  =  M'  {PC.  CC'a  +  £'2a>)  ; 

a~ 
and  also,  Mv  =  R,     M      u>  =  Rh  ; 

6 

hence  from  (11)  we  obtain  by  substitution 

G  =  — }  |>3  (A  -  r)  -(a-  rf  («2  +  2ha  +  hr)]  =  ~-3  {r  +  a)2  {2ah  -  a2  -  hr). 

8.  In  Ex.  7  find  at  what  point  of  the  rod  the  tendency  to  break  is  a  maximum. 
dG  I        2a  \ 

The  value  of  r  which  makes  — -  =  0  is  a  (  1  -  —  ) .  If  we  substitute  this  value 
dr  \        oh' 

&G  d2G 

of  r  in  G  and  in  — —, ,  we  find  that  G  is  positive  and  -~  negative  when  3A  >  a ; 
dr*  >  ar~ 

d2G 
also  G  is  negative  and  — j  positive  when  3A  <  «.     Hence  in  any  case  the  ten- 

/        2a\ 
dency  to  break  is  a  maximum  when  r  —  a  11—  —  1  . 

x2 


308     Free  Motion  of  Rigid  Body  parallel  to  Fixed  Plane. 

247.  Impulsive  Friction. — When  two  rough  surfaces 
collide,  the  investigation  of  what  takes  place  is,  in  general, 
somewhat  complicated.  We  must  regard  R  and  F,  the  im- 
pulses of  the  normal  reaction  and  friction,  as  variable  quan- 
tities, connected,  at  each  instant  of  the  impact,  by  linear 
equations  with  the  coexisting  values  of  the  velocities  of  rota- 
tion of  the  bodies  and  of  the  velocities  of  translation  of  their 
centres  of  inertia.  The  laws  which  regulate  the  impulse  of 
friction  may  then  be  stated  as  follows  : — 

(1)  The  direction  of  the  elementary  impulse  dF  due  to 
friction  is  opposite  to  that  of  the  slipping  of  the  point  of 
contact,  if  there  be  slipping ;  and  if  there  be  no  slipping,  is 
such  as  to  prevent  slipping. 

(2)  The  magnitude  of  dF  is,  if  possible,  just  sufficient  to 
prevent  slipping,  and  when  slipping  takes  place  dF  =  fxdR, 
ju  being  the  coefficient  of  dynamical  friction. 

The  equations  of  motion  for  impulses  (Art.  242)  show 
that  the  relative  normal  and  tangential  velocities  of  the  points 
of  the  bodies  in  contact  are,  at  each  instant,  of  the  form 
AR  +  BF  +  C,  where  A,  B,  and  C  are  constant  during  the 
impact. 

The  value  of  R  is  at  first  zero  ;  when  it  becomes  Rx  (at  the 
end  of  the  first  period  of  the  impact),  the  relative  normal 
velocity  is  zero ;  and  the  maximum  value  of  R,  which  it 
assumes  at  the  end  of  the  whole  impact,  is  (1  +  e)Rx. 

These  principles  afford  a  sufficient  number  of  equations  to 
determine  the  motion ;  and,  in  the  case  of  motion  parallel  to  a 
fixed  plane,  the  equations  are  always  soluble. 

If  the  bodies  which  collide  are  perfectly  rough,  the  relative 
tangential  velocity  of  the  colliding  points,  or  the  velocity  of 
slipping,  is  always  zero ;  and  when  R  =  Rx,  the  relative  nor- 
mal velocity  is  likewise  zero.  Hence  we  have  two  equations  to 
determine  Rx  and  the  corresponding  value  of  F.  At  the  end 
of  the  impact  R  =  (1  +  e)Rx ;  and  the  relative  tangential  velocity 
being  still  zero,  the  corresponding  value  of  F  can  be  de- 
termined. 

If  the  bodies  slip  on  each  other  in  the  same  direction  during 
the  whole  of  the  impact,  dF  is  always  equal  to  /mdR ;  hence 
F=  fxR  throughout.  Rx  is  then  determined  from  the  equation 
expressing  that  the  relative  normal  velocity  is  zero  ;  and  the 


Examples.  309 

final  values  of  R  and  F,  which  determine  the  motion  after  the 
impact,  are  (1  +e)El     and  /u(l  +  e)Rx. 

For  a  discussion  of  the  problem  in  more  complicated  cases 
the  reader  is  referred  to  Routh,  Rigid  Dynamics. 

If  a  sphere  impinges  against  a  fixed  surface,  or  if  two 
spheres  collide  with  each  other,  the  relative  tangential  velo- 
city v  depends  upon  the  velocities  of  rotation  of  the  spheres, 
and  the  velocities  of  their  centres  parallel  to  the  common 
tangent.  It  is  therefore  independent  of  the  normal  reaction, 
and  the  relative  normal  velocity  in  like  manner  is  independent 
of  the  friction.  In  this  case,  if  v  become  zero  it  must  re- 
main zero,  as  friction  cannot  initiate  a  relative  tangential 
velocity  in  its  own  line  of  direction.  Hence  v  must  be  either 
zero  at  the  end  of  the  impact,  or  in  the  same  direction  as  at 
the  beginning.  Moreover,  the  value  of  Rx  is  independent  of 
F.  The  problem  is,  therefore,  reducible  to  one  of  the  two 
cases  treated  above. 

If  we  assume  at  first  that  there  is  no  slipping,  and  obtain 
the  final  value  of  F  on  this  hypothesis,  the  solution  is  correct, 
provided  the  value  of  .Fso  obtained  does  not  exceed  fx(l  +  e)Ri. 
If  this  value  of  i^does  exceed  ju(1  +  e)Rl9  then  slipping  takes 
place  in  the  same  direction  throughout  the  impact,  and  the 
final  value  of  F  which  determines  the  subsequent  motion  is 

Examples. 

1.  A  box,  placed  on  a  rough  horizontal  table,  carries  two  vertical  rods  which 
support  a  horizontal  rod  from  which  a  mass  m  is  suspended.  A  fine  string, 
fastened  to  the  box,  and  passing  over  a  pulley  at  the  edge  of  the  table,  is 
attached  to  a  mass  M '  which,  when  set  in  motion,  causes  the  box  and  suspended 
mass  m  to  move  with  a  uniform  velocity.  The  string  which  supports  m  is  now 
cut,  and  m  falls  into  the  box.  If  its  velocity  after  m  has  struck  it  be  equal 
to  its  original  velocity,  and  if  the  friction  on  the  axle  of  the  pulley  be  neglected, 
show  that  the  coefficients  of  impulsive  and  continuous  friction  are  equal. 

Let  M be  the  mass  of  the  box  and  frame-work,  v'  its  original  velocity,  /j.  the 
coefficient  of  dynamical  friction,  R  the  impulse  of  the  normal  reaction,  and  F 
the  impulse  of  the  friction,  developed  between  the  table  and  box  when  the  latter 
is  struck  by  m.  Since  the  box  originally  moves  with  a  constant  velocity,  we 
have  M'g  =  /x  (M  +  m)g.  After  the  string  supporting  m  is  cut,  the  box  is  acted 
on  by  an  acceleration/,  during  the  time  t,  in  which  m  is  falling.  If  I  ho  the 
moment  of  inertia  of  the  pulley,  and  a  its  radius,  /  is  given  by  the  equation 


(M'  +  M+j\f=(M'-fiM)gi=iimg. 


(a) 


310     Free  Motion  of  Rigid  Body  parallel  to  Fixed  Plane. 

The  velocity  of  the  box  when  struck  by  m  is  v'  +  ft.      Hence  its  velocity  v 
after  the  impact  is  given  by  the  equation 


(m  +  m  +  M'  +  -2)  v  =  (m+  M'  +  -J  (v'  +  ft)  +  mv'  -  F. 


(*) 


If  v  =  */,  from  (b)  we  get  I M  +  M'  +—)ft  =  F;   this,  by  (a),  is  reduced  to 

F '■=  fxmgt',  but  R  =  mgt,  and  therefore  F=  fiR. 

This  example  is  a  description  of  the  experiment  by  which  Morin  showed 
that  impulsive  and  continuous  friction  obey  the  same  law,  and  have  the  same 
coefficient. 

If  the  friction  on  the  axle  of  the  pulley  were  taken  into  account,  the  terms 
arising  from  thence  in  the  above  equations  would  each  contain  as  a  factor  the 

quantity  -,  where  o  is  the  radius  of  the  axle.      But  as  o  is  very  small  compared 

with  a,  these  terms  may  be  neglected. 

2.  A  sphere,  rotating  with  an  angular  velocity  n  round  a  horizontal  axis 
at  right  angles  to  the  plane  of  the  trajectory  of  its  centre,  impinges  on  a  perfectly 
rough  horizontal  plane :  find  the  motion  immediately  after  impact. 

Suppose  the  sphere  is  moving  from  left  to  right  before  impact  with  a  velocity  V, 
whose  direction  makes  an  angle  i  with  the  plane  of  the  horizon.  Let  co  be  the 
angular  velocity  in  the  direction  of  the  motion  of  the  hands  of  a  watch,  and 
v  the  horizontal  velocity  of  the  centre  at  the  instant  after  impact.  F  being 
the  impulse  arising  from  friction,  the  equations  of  motion  are 

Mv  =  MV  cos  i  +  F, 
i-Maia,  =  %Ma2Cl-aF. 

The  geometrical  condition  for  no  slipping  is 

v  —  aco  =  0  ; 

F 

whence  —  =  -${Vcosi-ati), 

M 

v  =  aw  =  fFcosi'+f  an. 

If  V  cos  i  =  an,  no  impulsive  friction  is  called  into  play.  If  V  cos  i  >  an,  the 
horizontal  velocity  of  the  centre  of  the  sphere  is  diminished,  and  the  sphere  re- 
bounds at  a  greater  angle  than  if  there  were  no  friction.  If  V  cos  i  <  an  the 
horizontal  velocity  of  the  sphere  is  increased,  and  the  sphere  rebounds  at  a 
smaller  angle  than  if  there  were  no  friction.  In  this  case  friction  accelerates  the 
horizontal  velocity  of  the  centre  of  the  sphere. 

If  n  is  opposite  in  direction  to  the  motion  of  the  hands  of  a  watch, 

v  =  fVcosi  —  fan. 

The  velocity  of  the  centre  of  the  sphere  along  the  horizontal  line  is  dimi- 
nished, and  the  sphere  will  rebound  at  a  greater  angle  than  if  there  were  no 
friction.  If  5  V  cos  i=2aQ,  the  sphere  will  rebound  vertically.  If  2an>5Vcosi 
the  sphere  will  hop  back.  This  explains  the  effect  of  slow  under-cut  in  tennis. 
The  numerical  factors  for  a  tennis  ball  may  of  course  be  different  from  those 
given  above. 


Rolling  and  Twisting  Friction.  311 

The  magnitude  of  the  total  normal  reaction  between  the  sphere  and  the  plane 

,  .  ,  2(Fcosi-  ad)     . 

is  M(l  +  e)  Tsin i.     Ilence,  m  any  case  m  which  /x>  — rj=~. — :,  the  pre- 

/  ( 1  -J-  cj  v  sin  i 

ceding  investigation  holds  good,  even  though  the  plane  he  not  perfectly  rough. 
If  n  be  counter-clockwise  its  sign  must  be  changed  in  the  above  expression  for 
the  limiting  value  of  /j.. 

3.  If  the  plane  in  the  last  example  be  imperfectly  rough,  so  that  the  impul- 
sive friction  is  not  sufficient  to  destroy  the  whole  tangential  velocity  of  the  point 
of  contact  of  the  sphere  with  the  plane,  determine  the  motion. 
The  equations  are,  if  V  cos  i>«n, 

Mv  =  MVcos  i  -  /jl(1  +  c)MVsmi, 

fMarw  =  i-Mara  +  p(l  +  e)  MVa  sin/. 

The  sign  of  /j.  must  be  changed  in  these  equations  if  V  cosi  <  an,  and  the 
sign  of  n  if  its  direction  be  counter-clockwise. 

248.  Rolling  and  Twisting  Friction. — In  questions 
relating  to  friction,  if  great  accuracy  be  required  in  the 
determination  of  the  motion,  it  is  necessary  to  take  into 
account  not  only  the  tangential  force  of  friction,  but  also 
what  is  called  the  couple  of  rolling  friction,  which  is  a  couple 
having  for  its  axis  the  tangent  to  the  rough  surface  round 
which  the  body  is  rotating.  Its  maximum  value  is  the 
normal  pressure  multiplied  by  a  linear  constant,  and  is 
generally  small  in  amount,  so  that  in  solving  questions  con- 
nected with  friction  this  couple  is  usually  neglected.  The 
direction  in  which  the  couple  of  rolling  friction  tends  to  turn 
the  body  is  opposite  to  that  in  which  it  is  actually  rotating. 
If  the  body  be  not  actually  rotating,  but  be  acted  on  by  forces 
tending  to  make  it  rotate,  the  couple  of  rolling  friction  tends 
to  prevent  rotation  round  a  common  tangent  to  the  two  rough 
surfaces,  j 

If  the  surfaces  have  a  relative  angular  velocity  about  the 
common  normal,  then,  besides  the  tangential  force  of  friction, 
and  the  couple  of  rolling  friction,  there  is  also  a  couple, 
having  the  normal  as  its  axis,  called  the  couple  of  twisting 
friction.     This  couple  likewise  is  usually  small  in  amount. 

Examples. 

1.  Taking  into  account  the  couple  of  rolling  friction,  and  supposing  the 
motion  to  be  still  pure  rolling,  determine  in  Ex.  9,  Art.  245,  the  amount  of 
friction  brought  into  play,  and  the  angular  velocity  in  any  position. 


312     Free  Motion  of  Rigid  Body  parallel  to  Fixed  Plane. 

Let  R  be  the  normal  reaction  between  the  plate  and  circle  at  any  time,  and 
fR  the  couple  of  rolling  friction  ;  then  the  equations  of  rotation  of  the  plate  are 

du>  d20 

\mc&  —  =  Fa  -fR,  m (b  -  a)  — -  =  F- mg  sin  0. 

Alsoi2  =  m{gcos6+  (b-a)02},  and  (b-a)6  =  -aa>. 

Hence,  putting    -  =  v,    F=lm  J<7sin0  +  2vgoosB  +  2v{b-a)e~  J  ; 

consequently  we  obtain 

§{b-a)  —  =  AgcosO  +  (b-a)eA  -  g  sin 0. 

If  we  change  the  independent  variable  by  means  of  the  symbolic  equation 

d        .  d  q 

—  =  0  — ,  and  put  r1 —  =  n,  we  get 
dt         d0  *     b  -  a 

—  J  02  j   =  a  j  n^v  cos  0  _  sin  fl)  +  y02 

The  solution  of  this  differential  equation  is  of  the  form 

02  =  ce*"9  +  Dcos0  +  F  sin  0, 

where  C  is  an  arbitrary  constant.     Determining  the  constants  D  and  F,  we 
obtain 

02  =  c£v0  +  _i!!_   ( (3  _  4„2)  cos  e+7u  sin  e  )  . 
9+16j/2(  ) 

If  0o  be  the  initial  value  of  0,  we  have,  since  0o  =  0, 


0=-  9  3    (3  -  4i/2)  cos  0o  +  7?  sin  0O 

When  0  is  determined,  <w  can  be  found  by  the  equation  aa>  =  -  (b  -  a)0. 

2.  A  circular  plate  is  projected  along  a  rough  horizontal  plane,  with  an 
initial  velocity  V  of  translation,  and  an  angular  velocity  n,  round  an  axis  through 
its  centre,  at  right  angles  to  its  plane.  Determine  the  motion,  neglecting  the 
couple  of  rolling  friction. 

Let  o>  denote  the  angular  velocity  — ,  and  v  the  velocity  of  the  centre,  at  any 

dt 
time,  and  let  x,  the  horizontal  coordinate  of  the  centre,  be  measured  in  the  direc- 
tion of  V,  as  in  the  figure ;  then  the  whole  velocity  of  P  is  v  -  aw,  where  a  is 
the  radius  of  the  plate. 


X  P  X 

Different  phenomena  present  themselves  according  to  the  values  of  Fand  A 
(1)  ft  positive,  and  V>  aCi. 


Examples.  313 

Since  V-  a&  is  positive,  P  begins  to  slip  along  PX  ;  therefore  F=  fxMg,  and 
the  equations  of  motion  are 

M  _?  =  -  fiMff,     \Ma~  —  =  iiMgn. 
dt~  at 

V-  an 
Pure  rolling  commences  when    v  —  aw  =  0,  i.  e.  at  a  time  to  equal  to  — ; 

then  aw  =  v  =  |  F+  £  aft. 

The  equations  for  the  subsequent  motion  are 

v  =  aw,     Mp-=-F,    \Ma*%  =  Fa, 
dt  'at 

where  F  is  the  amount  of  friction  brought  into  play. 

Hence  F=0,  and  the  disk  will  roll  on  with  a  constant  velocity  of  rotation 
round  the  instantaneous  axis. 

(2)  n  positive  as  before,  Y<aCl. 

Since  V—  aco  is  negative,  P  commences  by  slipping  back  towards  X'. 

The  equations  given  above  must  in  this  case  be  modified  by  changing  the 
sign  of  ix.  The  initial  velocity  of  translation  of  the, centre  is,  in  this  case, 
increased. 

(3)  Initial  angular  velocity  negative  and  equal  to  -  n. 

Here  we  must  change  the  sign  of  Cl  in  the  equations  of  case  (1). 

If  an>  2  V,  both  v  and  w  will  be  negative,  that  is,  the  motion  of  translation 
of  the  centre  will  be  in  the  direction  opposite  to  that  originally  imparted,  and 
the  rotation  will  be  in  the  same  direction  as  the  initial  rotation. 

3.  Discuss  the  same  problem,  taking  into  account  the  couple  of  rolling 
friction. 

Here  we  have 

M^.  =  -  fiMff,     \Ma^  =  nMga  -fMg. 
at  at 

f 
Putting  -  =  v,  we  find  then  that  pure  rolling  commences  when 

V-aO, 


{Zix-2v)g 


At  this  instant  v  =  — — — — —  =  t>o- 

3/j.  —  2j/ 

After  this  the  equations  of  motion  become 

*£?--■*•  '"i-*"* 

along  with  v  =  aw  ;  whence  F=  *vMg. 

This  expression  shows  that  the  friction  brought  into  play  varies  inversely  as 
the  radius  of  the  plate,  provided  its  mass  be  constant. 
The  plate  will  come  to  rest  at  a  time 

,,  _  3f0 
"  2r9* 
where  f  is  counted  from  the  instant  when  pure  rolling  begins. 


314     Free  Motion  of  Rigid  Body  parallel  to  Fixed  Plane. 

In  order  that  the  motion  should  hecome  pure  rolling  it  is  necessary  that 
M  >  %v. 

The  student  will  have  no  difficulty  in  investigating  cases  (2)  and  (3)  of  Ex.  2, 
when  the  couple  of  rolling  friction  is  taken  into  account. 

4.  A  sphere  is  projected  down  a  rough  inclined  plane,  along  a  line  of 
greatest  slope  of  the  plane.  The  sphere  has  an  initial  velocity  of  rotation  round 
a  horizontal  axis  parallel  to  the  inclined  plane  ;  determine  the  motion— (1)  neg- 
lecting the  couple  of  rolling  friction ;  (2)  taking  that  couple  into  account. 

Let  the  line  of  projection  be  the  axis  of  x,  and  let  x  positive  he  measured  to 
the  right,  and  a>,  the  angular  velocity,  be  in  the  direction  of  the  motion  of  the 
hands  of  a  watch.  Let  F"be  the  initial  velocity  of  translation  of  the  centre  of 
the  sphere,  and  D  the  initial  angular  velocity. 

(1)  The  equations  of  motion  are, 

~rd'2x      ,,     .     .      _       „  _,  nd(a     _ 
M-—  =  Mg  sin  *  -  F,      f  Ma2  —  =  Fa, 
dt*  at 

and  the  condition  for  pure  rolling  is 

v  —  au  =  0. 
If  ^0  be  the  time  at  which  pure  roiling  begins,  then 

2{V-aD)  2  (an-  V) 


(7ucosi-  2sini)g'  (7/xcos  i+  2  sin  i)g* 

according  as  V>  aD,  or  aD  >  V,  where  u  is  the  coefficient  of  dynamical  friction. 
If  V—  aD  >  0,  we  must  have  7,u  cos  i  >  2  sin  i  in  order  that  pure  rolling  should 
be  attainable.     If  V  —  aD.  =  0,  pure  rolling  will  continue,  provided  7fi'  cos  i 

>  2  sini  (where  /x  is  the  coefficient  of  statical  friction).  If  V -  aD.  <  0,  pure 
rolling  will  be  reached  necessarily,  and  will  then  continue,  provided  7/*'  cos  i 

>  2  sinr. 

If  vo  and  w0  be  the  values  of  v  and  «  when  pure  rolling  is  attained, 

bu.  V  cos  i  —  2  (sin  i  —  u  cos  i)  aD. 

aw0  =  v0= \  "  . — , 

7/i  cos  j-2sini 

bu  V  cos  i  +  2  (sin  i  +  u  cos  i)  aD 
Ifi  cos  %  +  2  sin  % 

according  as  V  —  aD  is  positive  or  negative.  It  may  be  observed  that  the 
equations  for  the  latter  case  can  be  obtained  from  those  for  the  former  by 
changing  the  sign  of  u.     After  pure  rolling  begins,  if  it  continues,  F=  f  Mg  sin 

v  =  aw  =  y  (t  —  to)  g  sin  i  +  aa>0. 

(2)  The  equations  of  motion  are 

M  —  =  Mg  sin  i  -  F,      f  Ma2  -^  =  Fa  -fMg  cos  i. 
ut  at 


Examples.  315 


Hence,  putting  -  =  v,  we  have,  when  V>  aQ.,  n  being  positive, 
a 

2(V-aa) 

tn  = 


{ 7fi  —  bv)  cos  i  —  2  sin  i)  g ' 

and  in  order  that  pure  rolling  may  be  possible,  l\i  -  bv  >  2  tan  i. 

b(n-v)  V  cos  i  -  2  (sin  t  -  ju  cos  i)  aH 
Again,  r0  =  «a>0  =  (7/1  _  5„)  cos  i  -  2  sin  *  ' 

and  at  any  time  after  pure  rolling  is  established, 

aw  =  «<w0  +  %9  (srn  *  _  v  cos  i)  (£  —  tQ). 

When  F  <  a  n,  the  equations  corresponding  to  this  case  are  obtained  from 
those  above  by  changing  the  sign  of  /*. 

If  the  initial  angular  velocity  be  negative,  and  equal  to  -  n,  the  equations  of 
motion  are 

M  — -  =  Mg  sin  i  —  /i  Mg  cos  i, 
dv 

tMcfi  —  =  dfiMg  cos  i  +/Mg  cosi, 
dt 

until  co  =  0.     This  takes  place  at  a  time  ti  given  by  the  equation 

2«fl 

bg  cos  i  (n  +  v)' 
Then 

_  la  n(sin  i  -  /x  cos  i)  +  5  F  cos  i  (/*  +  f ) 
5  cos  i  (n  +  v) 

After  this  o>  is  positive,  and 

2vx  n  {li-p)aa+(fi  +  v)V 

t0-h+  |(7/i_5„)cosj_2sini}#~  "  {fi  +  v){{7p-  5v)cosi-  2sini}/ 

^  _  „     5  (/z  +  v)  F  cos  i  4-  2  (sin  i  -  fx  cos  t)  aft 

and  v0  =  #w0  = .  tz        r~:         :      ~    :     :  • 

0  °      M  +  v  (7/i  -  5v)  cos  x  -  2  sin  t 

5.  A  number  of  spheres  are  projected  in  different  directions  with  different 
initial  velocities  along  a  rough  horizontal  plane  ;  find  the  path  of  their  common 
centre  of  inertia. 

Am.   A  series  of  parabolas,  and  finally  a  straight  line  (see  (1),  Ex.  2). 


316     Free  Motion  of  Rigid  Body  parallel  to  Fixed  Plane. 

6.  A  hollow  cylinder  filled  with  water  is  projected  without  initial  rotation 
in  a  direction  perpendicular  to  its  axis,  along  a  rough  horizontal  plane ;  deter- 
mine the  time  at  which  pure  rolling  hegins,  the  amount  of  friction  subsequently 
brought  into  play,  and  the  time  at  which  the  cylinder  comes  to  rest. 

Let  M  be  the  mass  of  the  cylinder  and  contained  water,  I  the  moment  of 
inertia  of  the  cylinder  round  its  central  axis,  a  the  radius  of  its  external  surface, 
fx  and /the  coefficients  of  sliding  and  rolling  friction,  Fthe  initial  velocity  of  the 
common  centre  of  inertia  G  of  the  cylinder  and  contained  water,  t\  the  time  at 
which  pure  rolling  begins,  F  the  friction  subsequently  brought  into  play,  vi  the 
velocity  of  the  point  G  at  the  time  h,  t2)  the  time  at  which  the  cylinder  comes 

/ 
to  rest.     Then,  putting  -  =  v,  we  find 
ci 


fi  +  A  (n  -  v)  g  1  +  A  /i  +  \(p-v)     '  vg 

where  a/=  Ma2.  As  A  increases,  F  increases,  and  so  in  general  does  vi,  whilst 
ti  diminishes,  and  to  in  every  case  remains  constant,  being  the  same  as  in  the 
case  of  a  solid  cylinder  (see  Ex.  3). 


(     317     ) 


CHAPTER  XL 

MOTION  OF  A  RIGID  BODY  IN  GENERAL. 

Section  I. — Kinematics. 

249.  Motion   of  a   Body    having    one    Point  fixed. — 

If  a  rigid  body  have  a  fixed  point,  a  spherical  surface  S  fixed 
in  the  body,  with  this  point  as  centre,  must  move  about  on  the 
surface  of  an  equal  concentric  sphere  fixed  in  space.  The 
position  in  space  of  S,  or  of  any  definite  great  circle  on  it, 
determines  that  of  the  body.  Hence  the  motion  of  a  body 
having  a  fixed  point  is  reducible  to  the  motion  of  a  spherical 
figure  on  a  sphere  fixed  in  space.  The  position  of  such  a 
figure  is  determined  by  the  positions  of  any  two  definite 
points  A  and  B  in  it.  If  the  points  A  and  B  move  into  new 
positions  A'  and  B\  arcs  of  great  circles  bisecting  AA'  and 
BB'  at  right  angles  will  meet  in  a  point  0,  and  the  angle 
AOA!  =  BOB';  but  the  great  circle  OA  can  be  moved  into 
the  position  OA'  by  turning  it  through  the  angle  AOA'  round 
the  axis  CO  (C  being  the  centre  of  the  sphere) ;  and  since 
AOA'  =  BOB ',  the  same  rotation  brings  OB  into  the  position 
OB'.  Hence  a  rotation  round  OC  brings  the  spherical  figure, 
of  which  A  and  B  are  definite  points,  from  the  first  position 
into  the  second.  The  point  0  is  called  the  pole  of  rotation 
(Differential  Calculus,  Art.  300). 

Consequently,  a  rigid  body  having  a  point  fixed  can  be  moved 
from  any  one  position  into  any  other  by  a  rotation  round  an 
axis  through  the  point. 

250.  Composition  of  Rotations  round  Axes  meet- 
ing in  a  Point. — If  a  body  receive  rotational  displacements 
round  two  axes  fixed  in  space,  passing  through  the  same 
point,  the  resultant  displacement  may  be  effected  by  a  rota- 
tion round  a  single  axis. 

If  the  displacements  be  infinitely  small,  it  appears,  as  in 
Article  220,  that  the  order  in  which  they  are  effected  is  in- 


318  Kinematics  of  a  Rigid  Body. 

different,  and  also  that  it  is  indifferent  whether  the  axes  be 
fixed  in  space  or  be  axes  fixed  in  the  body,  whose  positions  at 
the  commencement  of  the  infinitely  small  motion  coincide 
with  those  of  the  axes  fixed  in  space.  If  the  two  displace- 
ments be  regarded  as  simultaneous,  the  resultant  rotation  is 
the  actual  motion  of  the  body.     Hence  we  see  that — 

A  velocity  of  rotation  round  a  single  axis  is  equivalent  to 
velocities  of  rotation  round  two  axes  meeting  the  axis  of  the 
resultant  rotation  in  the  same  point. 

Beiny  given  the  velocities  of  rotation  of  a  rigid  body  round 
two  axes  meeting  in  a  point,  to  determine  the  velocity  of  the  re- 
sultant rotation  and  the  position  of  its  axis. 

Let  OA  and  OB  be  the  axes  of  the  component  rotations, 
and  R  a  point  on  the  axis  of 
the  resultant  rotation.     As  R  is 
at  rest  during  the  motion,  its  , 

displacement  from  the  rotation  / 

round   OA  must  be  equal  and  / 

opposite  to  that  from  the  rota-         / 
tion   round    OB.       Hence   the       / 

circles  passing  through  R,  and     /    

having   their   planes    at    right  jf 

angles  to  OA  and  OB,  and  their  \7~~       A  ,/r 

centres  on  those  lines,  touch  ati£.         ~~~ " 

Hence  OA,  OB,  and  OR  lie  in  the  same  plane.  This  appears 
readily  from  the  fact,  that  if  two  small  circles  of  a  sphere 
touch,  the  arc  of  a  great  circle  joining  their  poles  passes 
through  the  point  of  contact.  Again,  AR  multiplied  by  the 
angular  velocity  round  OA  is  equal  and  opposite  to  BR  mul- 
tiplied by  the  angular  velocity  round  OB.  If  these  angular 
velocities  be  denoted  by  a  and  |3,  we  have 

a  _  P 

sin  B  OR  sin  A  OR' 
To  find  a,  the  angular  velocity  of  the  resultant  rotation, 
consider  the  motion  of  A.  It  is  unaffected  by  the  rotation 
round  OA,  and  may  be  regarded  indifferently,  as  rotating 
round  OB  with  angular  velocity  j3,  or  as  rotating  round  OR 
with  angular  velocity  w. 


Composition  of  Rotations  round  Axes  meeting  in  a  Point.     319 

If  perpendiculars  AP  and  AQ  be  let  fall  on  OB  and  OR, 
we  have  then  AP. [3  =  AQ.w.     Hence 

(x)  (5  a 


sin  A  OB     sin  A  OR      sin  BOR 

Hence,  finally — The  axis  of  the  resultant  rotation  lies  in  the 
same  plane  as  the  axes  of  the  component  rotations,  and  makes 
with  each  an  angle  whose  sine  is  proportional  to  the  velocity  of 
rotation  round  the  other  ;  and  the  velocity  of  the  resultant  rota- 
tion is  proportional  to  the  sine  of  the  angle  beticeen  the  axes  of  the 
component  rotations. 

Accordingly,  velocities  of  rotation  are  compounded  in 
precisely  the  same  manner  as  velocities  of  translation,  or  as 
forces  meeting  in  a  point. 

By  reversing  the  reasoning  above,  it  can  be  shown  that  a 
point  R,  taken  as  above,  remains  at  rest  under  the  influence 
of  two  velocities  of  rotation  round  OA  and  OB ;  whence  we 
have  an  independent  proof,  that  infinitely  small  rotations 
round  two  intersecting  axes  are  equivalent  to  a  single  one 
round  an  axis  lying  in  the  plane  of  the  two  former,  and 
passing  through  their  point  of  intersection. 

We  have  already  seen,  Article  221,  that  velocities  of 
rotation  round  parallel  axes  are  compounded  in  the  same  way 
as  parallel  forces.  Hence,  in  general —  Velocities  of  rotation 
are  compounded  like  forces,  whose  directions  coincide  with  the 
axes  of  rotation,  and  whose  magnitudes  are  proportional  to  the 
velocities  of  rotation. 

The  attention  of  the  reader  has  been  directed  in  Article 
221  to  the  algebraical  signs  of  velocities  of  rotation.  In 
addition  to  what  was  there  stated,  it  may  be  observed, 
that  the  axis  of  a  rotation  may  be  made  to  represent  the 
rotation  both  in  magnitude  and  direction.  In  this  case  the 
axis  is  drawn  so  that  the  rotation  round  it  is  always  positive. 
For  example,  instead  of  speaking  of  a  negative  rotation 
round  the  axis  of  X,  we  may  designate  it  simply  as  a  ro- 
tation round  the  axis  of  X  negative.  When  the  axis  of  a 
rotation  determines  the  direction  of  the  rotation,  the  latter  is 
always  understood  to  be  in  the  positive  direction  round  this 
axis,  that  is,  according  to  the  convention,  counter-clockwise. 


320  Kinematics  of  a  Rigid  Body. 

When  rotations  are  compounded  by  means  of  their  axes,  like 
forces,  the  direction  of  the  axis  determines  in  this  way  the 
direction  of  the  rotation. 

For  example,  rotations  a>i,  w2,  w3  round  three  rectangular 
axes  produce  a  resultant  rotation  w  which  is  always  positive  ; 

but  the  direction  of  its  axis  is  determined  by  — ,  — ,  — 3,  the 

(1)         It)         (s) 

cosines  of  the  angles  made  with  the  coordinate  axes  ;  and 
these  again  depend  on  the  signs  of  wi,  o>2,  and  wz,  as  well  as 
on  their  magnitudes. 

251.  Geometrical  representation  of  the  Motion  of 
a  Body  having  a  Fixed  Point. — When  a  body  has  a  fixed 
point,  its  motion  may  be  represented  in  a  manner  analogous 
to  that  mentioned  in  Article  225.  In  the  present  case  the 
curves  which  correspond  to  the  space  centrode  and  the  body 
centrode  are  spherical  curves  lying  on  the  surface  of  the  same 
sphere. 

The  motion  of  the  body  is  represented  by  the  rolling  of  a 
cone  fixed  in  the  body  on  a  cone  fixed  in  space  (see  Differen- 
tial Calculus,  Article  301). 

252.  Motion  of  a  Body  which  is  entirely  Free. — 
A  rigid  body  can  be  moved  from  any  one  position  into  any  other 
by  a  motion  of  translation,  combined  with  a  motion  of  rotation 
round  an  axis  through  any  arbitrary  point  A  of  the  body. 

Let  Ax,  A2  be  the  two  positions  in  space  occupied  by  A 
in  the  different  positions  of  the  body.  Give  to  every  point 
of  the  body  a  motion  equal  and  parallel  to  AiA2:  this  brings 
A  into  the  required  position,  and  a  rotation  round  an  axis 
through  A  will  then  (Article  249)  complete  the  body's 
change  of  place. 

If  two  positions  of  a  body  in  motion  are  infinitely  near 
each  other,  any  infinitely  small  displacements,  by  which  it 
can  be  moved  from  the  first  of  these  positions  to  the  second, 
may  be  regarded  as  the  actual  motion  of  the  body. 

The  actual  motion  of  a  rigid  body  during  an  infinitely 
short  time  is,  therefore,  a  motion  of  translation  together  with 
a  motion  of  rotation  round  an  axis  through  any  arbitrary 
point  of  the  body. 

The  initial  and  final  positions  of  a  body  being  given,  the  mag- 
nitude of  the  rotation,  which  is  required  to  make  it  pass  from  one  to 


Motion  of  a  Body  having  a  Fixed  Point.  321 

the  other,  and  the  direction  of  its  axis  are  determined ;  but  the 
motion  of  translation  varies  according  to  the  point  through 
which  the  axis  of  rotation  is  supposed  to  pass. 

First,  let  the  axis  of  rotation  be  supposed  to  pass  through 
a  point  A,  whose  initial  and  final  positions  are  Ax,  A2.  The 
motion  of  translation  AXA2  is  composed  of  two  parts — one^^I' 
in  the  direction  of  the  axis  of  rotation  through  A,  and  the 
other  AA2  at  right  angles  to  it.  By  means  of  the  first  a  defi- 
nite plane  section  of  the  body,  passing  through  A  and  at  right 
angles  to  the  axis  of  rotation,  is  moved  into  the  plane  in  space 
in  which  it  lies  in  its  final  position,  and  the  subsequent  motion 
of  the  body  is  therefore  parallel  to  this  plane.  If,  now,  the 
axis  of  rotation  be  regarded  as  passing  through  another  point 
B  of  the  body,  whose  initial  and  final  positions  are  Bx,  B2,  we 
can  suppose  the  translation  BXB2  made  up  of  two  parts — one, 
BXB\  equal  and  parallel  to  AXA ;  the  other,  B'B2,  which 
depends  on  the  position  of  the  point.  BxBf  brings  the  body 
into  the  same  position  as  AXA.  Hence,  a  translation  BfB> 
and  a  rotation  round  an  axis  through  B  are  equivalent  to 
an  equal  rotation  round  a  parallel  axis  through  A  and  a 
translation  A 'A,  (Art.  219).  The  translation  BXB2  is  the 
resultant  of  BXB'  and  B,B2;  AXA2  is  the  resultant  of  AXA ' 
and  AA2;  BxBf  is  equal  and  parallel  to  AXA;  but  B'B2  is 
not  in  general  either  equal  or  parallel  to  AA2. 

253.  Analytical  Treatment  of  the  Motion  of  a 
Body  having  a  Fixed  Point. — Suppose  three  rectangu- 
lar axes  fixed  in  the  body  passing  through  a  point  0 ;  and 
three  others  fixed  in  space,  which  at  the  beginning  of  the 
motion  coincide  with  the  former.  Let  the  coordinates  of 
any  point  of  the  body  referred  to  the  former  be  ?,  *?,  £,  and 
referred  to  the  latter,  ,r,  y,  z.  Let  au  a2y  a3;  bx,  b\,  b:i;  cls 
ciy  cz  be  the  cosines  of  the  angles  which  £,  »j,  £  make  with 
x,  y,  z,  respectively ;  and  let  the  angles  themselves  be  al9  a2, 
<*3 ;  /3i,  /32,  ^s ;  yi,  y2,  y& 

If  the  point  0  be  fixed,  we  have  at  any  instant 

x  =  alIi+blti  +  c1Zi  g  =  a2^+b2i]+c2Zy  z  =  (h%  +  b9ri  +  e3Z' 

If  at  this  instant  any  other  point  of  the  body  besides  0 
occupy  the  same  position  in  space  as  at  the  beginning  of  the 


322 


Kinematics  of  a  Rigid  Body. 


motion,  for  this  point,  a?  =  ?,  y  =  t»,  %  =  ?,  and  therefore  we 
should  have 

ax  -  1         £>t  ct 

tfo  ^2-1  <?2  =  0. 

#3  ^3  c3-  1 

But  it  is  easy  to  see  that  this  condition  is  fulfilled,  for 
putting 

tfi     bx     ^i 

a2     b2     Co 

Ch      O-i      c6 


=  A, 


and  denoting  the  former  determinant  by  A',  we  have,  by  the 
multiplication  of  the  determinants, 

AA'  =  -  A',  and  therefore  A'  =  0. 

Hence  we  conclude  that  if  a  rigid  body  have  a  fixed  point, 
any  two  positions  have  a  line  in  common. 
Again, 

dx  =  £dai  +  r)dbx  +  Zdd,     dy  =  %da2  +  r\db2  +  £dc2,    ' 

dz  =  £da3  +  rjdb3  +  %dcz ; 

but  since,  at  the  beginning  of  the  motion,  £,  rj,  £  coincide 
with  x,  y,  z,  we  have  at  that  instant 

ax  =  cos  ot ;  .*.  dch  =  -  sin  axdax  =  0,   since  ax  =  0. 
In  like  manner  db2  =  0,     dc6  -  0  ; 

also  axbx  +  a2b2  +  aAb3  =  0. 

Differentiating,  and  remembering  that  initially 

ax  =  1,     a2  =  0,     #3  =  0,     £>i  =  0,     b2  =  1,     63  =  0, 
we  have  dbx  +  da2  =  0. 

In  like  manner     dcx  +  da3  =  0,     dbz  +  dc2=0. 


Motion  of  a  Body  having  a  Fixed  Point.  323 

Let  now  da2  =  dip,     db^  =  dO,     dcy  =  d$  ; 

then  dx=-i)d\p+  Z>d$  =  - ydxp  +  zd<p  \ 

dy  =  -  Z,dQ  +  %dxp  =  -  zdd  +  <ed$     .  (1) 

dz  =  -  %d<p  +  r)dO  =  -  ccdtp  +  ydd  ) 
But  a  rotation  dO  round  x  would  give  (Art.  222) 

d>/  =  -Zd9,     dz  =  r}d9; 
d(p  round  y  would  give 

dz  =  -  %d(p,     dx  =  Z>d<p  ; 
and  d\p  round  2  would  give 

dx  =  -  r]d<p,     dy  =  £d>p. 

Hence  the  most  general  infinitely  small  displacement  the 
body  can  take,  0  remaining  fixed,  is  equivalent  to  rotations 
round  any  three  rectangular  axes  through  0. 

Moreover,  from  the  values  of  dx,  dy,  dz,  given  above,  it 
appears  that  for  a  point  whose  coordinates  fulfil  the  condi- 
tions -tq  =  —  =  —  the  displacements  are  zero. 

Hence  the  three  rotations  dO,  dcp,  d\p,  round  the  axis  x,y,z, 
are  equivalent  to  a  single  rotation  round  an  axis  whose  posi- 
tion is  defined  by  these  equations.     If  we  put 

dO  =  dx  cos  A,     d(p  =  d%  cos  fx,     dip  =  dx  cos  v, 

where  dx  =  V^dO2  +  d§~  +  dip2, 

the  equations  of  the  fixed  axis  are 


cos  A      cos  fi     cos  v 
Also,  for  any  point  of  the  body, 

dx2  +  dy2  +  dz2  =  [(»?  cos  v  -  Z  cos  /m)2  +  (£  cos  A  -  £  cos  v)2 

+  (£  cos  f.i  -  ?j  cos  A)2]  dx2  =P2  dx2y 

if  p  be  the  perpendicular  from  the  point  on  the  fixed  axis. 
Hence  d\  is  the  magnitude  of  the  resultant  rotation. 

Y2 


324  Kinematics  of  a  Rigid  Body. 

Whence  infinitely  small  rotations,  and  therefore  velocities 
of  rotation,  are  compounded  like  forces  meeting  at  a  point. 

254.  Motion  of  a  Body  entirely  Free. — If  the  point 
of  intersection  of  the  axes  fixed  in  the  body  be  itself  in 
motion,  and  if  its  coordinates,  referred  to  axes  fixed  in  space, 
be  x\  y,  z'  ;  then,  for  any  point  xyz  of  the  body, 

x  =  x  +  «!$  +  bit)  +  dZ,     y  =  y  +  a.£  +  b2ri  +  c2Z, 

s  =  z  +  a3£  +  bzr\  +  CzZ; 
whence 

dx  =  dx  +  5  dax  +  r}dbi.  +  Z  dch  dy  =  dy'  +  %  da2  +  rj  db2  +  Z  dc2y 

dz  =  dz  +  %daz  +  ridbs  +  Zdc3. 

If  we  suppose  the  axes  of  £,  n,  Z  parallel  to  those  of  x,  y,  z 
at  the  beginning  of  the  motion,  we  get,  as  in  the  last  Article, 

dx  =  dx'  -  y\d\p  +  Zd(j)  =  dx'  -  (y  -  y)  dij/  +  (z  -  z')  d(p 

dy  =  dy--ZdO  +  &ty  =  dy'-{z-z)dd+{x-x')dil,  L  (2) 


! 


dz  =  dz'  -Z,d<j}  +  n  d6  =  dz'  -(x-  x')d^  +  (y  -  y)  dQ 

and  we  see  that — 

The  most  general  infinitely  small  displacement  which  a  rigid 
body  can  receive  consists  of  a  movement  of  translation,  and  a 
movement  of  rotation  round  an  axis  through  any  arbitrary  point 
of  the  body. 

Again,  whatever  be  the  point  through  which  the  axis  of  rota- 
tion is  supposed  to  pass,  the  direction  and  magnitude  of  the  rota- 
tion remain  unaltered. 

Suppose  two  points  x'y'z',  x"y"z",  successively  regarded  as 
the  points  through  which  the  axis  of  rotation  passes  ;  then, 

dx  =  dx"  -  (y  -  y")dx\,"  +  (z  -  z")  df; 
also  dx  =  dx"  -  [y  -  >/")  d^"  +  (z'  -  z")  df. 


Subtracting,  we  get 

dx  =  dx  -  (y  -  y')d$'  +  (z  -  z)d$"\ 


Velocity  of  any  Point  of  a  Body. 


325 


but  again, 

dx  =  dx  -  (y  -  y') dif/  +  (s  -  z) d<p\ 

Comparing  these,  we  see  that 

d\p'  =  dif/'\     dtj>'  =  dtf ' . 

In  like  manner  dd'  =  dQ";  hence  the  rotation  remains  un- 
altered in  magnitude  and  direction. 

255.  Velocity  of  any  Point  of  a  Body. — Infinitely 
small  displacements  divided  by  the  element  of  time  during 
which  they  are  effected  become  velocities.  If  the  axes  of 
x,  y,  z  be  three  rectangular  axes  fixed  in  space,  and  if  the 
velocities  of  rotation  round  parallel  axes  meeting  at  the  point 
x'y'z\  be  wx,  wy,  w2,  we  have,  from  equations  (2), 

dx      dx  /         /\       1 

dt      dt 


dy 
dt 


dy 
~dt 


{y-y)<*>*  +  (*- 

(S  -  JO  €*  +  (*- 


X    )(i)r 


dz  _dz__ 
~dt~~di~^X 


)  »>y  +  (y  -  y') 


X  )o>. 


(3) 


If  the  point  xfyz  be  fixed  in  space,  and  be  taken  for 
the  origin,  we  have 


dx 

dy 


wzy 


wxz    y- 


dz 

7t 


=    Wxy    -    (t)yX 


(4) 


If  we  suppose  the  axes  fixed  in  space  to  coincide  at  the 
instant  under  consideration  with  axes  fixed  in  the  body,  and 
if  the  angular  velocities  round  the  latter  be  wi,  wz,  a*,  we 
have  o)x  =  m,  <*>v  =  w2,  wz  =  w,.  Consequently,  if  S,  v,  2  be  the 
coordinates  of  any  point,  referred  to  axes  fixed  in  the  body, 


326 


Kinematics  of  a  Rigid  Body. 


and  if  u,  v,  w  be  the  components  of  its  velocity  parallel  to  these 
axes,  we  have 


u 


v  =  u)£ 

W  =  Wit] 


(I) 


K 


(5) 


Equations   (3),   (4),  and  (5)  hold  good  for  every  instant, 

whereas  the  equations  x  =  £,  &e.,  wx  =  w„  &e.,  —  =  u,  &c, 

(it 
hold  good  only  for  one  particular  instant. 

If  A,  jjl,  v  be  the  direction  cosines  of  a  definite  line  in  the 
body  referred  to  axes  parallel  to  fixed  directions  in  space,  we 
have,  as  an  immediate  consequence  of  (4), 


tfA 


dt 

(DyV    - 

W2jU 

dn 

dt 

= 

wzX  - 

wrv 

i 

dv 
dt 

= 

WxjU  - 

U)y\ 

>■  (6) 


The  motion  of  a  body  relative  to  the  space  in  which  it  is 
moving  is  unaltered  if  we  attribute  to  the  latter  the  motion 
of  the  body  reversed,  and  suppose  the  body  itself  to  be  at  rest. 
Hence,  if  /,  m,  n  be  the  direction  cosines  of  a  line  fixed  in 
space  referred  to  body  axes,  we  may  regard  the  latter  as  fixed 
in  space,  and  the  line  Imn  as  moving  round  them  with 
angular  velocities  -  an,  -  a>2,  -  a>3.  Accordingly,  from  (6), 
we  have 

dl  -\ 

=  —  him.     4-  num. 

dt 


=  —  u)zn    +  oj-jn 


dm 
dt 

dn 

Jt 


=  -  h)Zl     +  d)\n     y  • 


-—  =  -  wim  +  w2/ 


(7) 


Acceleration  of  Rotation.  327 

\  256.  Acceleration  of  Rotation. — If  wl5  w2,  w3,  be  the 

angular  velocities  round  three  rectangular  axes,  OA,  OB, 
00  fixed  in  the  body,  and  wx,  wy,  wz  the  velocities  round  axes 
OX,  OY,  OZ  fixed  in  space  ;  and  if  at  any  instant  we  suppose 
OX,  OY,  OZ  to  coincide  with  the  positions  occupied  at  the 
instant  by  OA,  OB,  OC,  then  not  only  is  on  equal  to  wx,  w* 
to  wu,  and  o>3  to  wz,  but  also 


du>\ 

dwx 

dwz 

d(M)y 

dwz      dw% 

"dt 

=  Ht9 

dt 

^Hf' 

lit  =~di 

This  may  be  proved  as  follows  : — 

Let  w  be  the  velocity  of  rotation  round  a  line  fixed  in  the 
body,  which  passes  through  0,  and  makes  angles  with  the 
axes  OX,  OY,  OZ,  whose  direction  cosines  are  A,  ju,  v ;  then 

(A)  =    Wz\    +    WyfJ.    +    U)ZV   ', 

.       .  da)      dwx  A      dwy        du)z 

therefore  — -  =  —  A  +  —  n  +  —  v 

dt        dt  dt  dt 

dX  dfx  dv 

+  Ux-77  +  WyTT    +    ^3  -JT • 

dt  dt  dt 


Hence,  by  (6),  =  A  —  +  ^  —  +  v  — .  (8) 


This  equation  shows  that  the  acceleration  of  rotation 
round  a  line  is  the  differential  coefficient,  with  respect  to  the 
time,  of  the  angular  velocity  round  the  same  line  even  though 
it  is  in  motion,  provided  it  be  fixed  in  the  body. 

Thus,  in  particular,  we  have  in  the  case  supposed  above, 

du)i      d(x)x      du)2      dwy      dw3  ^  diDz  ,qv 

lU  =  Hi'     dt  ==  It'     lit "~~~  lit'  ^  ' 

The  same  may  be  proved  geometrically  as  follows : — 

The  body  at  any  instant  is  rotating  round  a  certain  axis 
with  an  angular  velocity  w.  Draw  a  line  through  the  fixed 
origin  in  the  direction  of  the  instantaneous  axis,  and  measure 
off  on  it  a  portion  01,  proportional  to  o) ;  then  the  projec- 


328  Kinematics  of  a  Rigid  Body. 

tions  of  this  line  on  the  axes  fixed  in  space  represent  wXi  wyf  wz ; 
and  its  projections  on  the  axes  fixed  in  the  body  represent 
&>i,  w2,  ws-  At  the  next  instant  the  body  is  rotating  round 
another  line  with  a  velocity  &>',  represented  by  OT,  and  the 
projections  of  OT  represent  w/,  w/,  «/;  oj/,  oj2',  w3'.  But  the 
projection  of  01'  is  equal  to  the  sum  of  the  projections  of 
01  and  IT.     Hence 

dwx  =  wx'  -  h)x  =  projection  of  IT  on  axis  of  x  fixed  in  space, 
d(t>i  =  wi  -  (1)1  =  projection  of  IT  on  axis  of  ?  fixed  in  the  body. 

At  the  first  instant  the  axes  of  x 
and  S  coincide ;  and  at  the  next  the 
two  projections  of  IT  differ  only  by  a 
quantity  infinitely  small  compared  with 
ITy  which  is  itself  infinitely  small  of 
the  first  order.  Hence  dux  and  dwv 
differ  by  an  infinitely  small  quantity 
of  the  second  order ; 

dwi      dwx      dwz      dd)y      dw% 
~dt==~aT'     ~dt=~dii     ~dt 

A  line  passing  through  0  parallel  to  IT  is  called  the  axis 
of  angular  acceleration.     If  we  put  — —  =  <bXi  &c,  the  magni- 

Civ 

tude  of  the  resultant  angular  acceleration  is  <y(ayx  +  d>y2  +  d>~2), 
as  it  is  the  resultant  of  the  three  accelerations  wx,  wy,  and  cbz. 

257.  Accelerations  of  a  Point,  parallel  to  three 
Axes  fixed  in  the  Body. — If  u,  v,  w  be  the  velocities  of  a 
point  parallel  to  axes  fixed  in  the  body,  its  velocity-component 
Vy  along  a  line  whose  direction  cosines  referred  to  these  axes 
are  /,  m,  n,  is  ul  +  vm  +  ten. 

If  we  suppose  this  latter  line  fixed  in  space,  the  accelera- 

dV 
tion  of  the  point  parallel  to  it  is  — -,  and  we  have 

at 

dV  7du  dv  dw  dl  dm  dn 
-jr  =  I  -77  +  m—  +  n  —  +  u  —  +  v  —  +  w  -—. 
dt  dt         dt         dt         dt         dt         dt 


Complete  Determination  of  the  Motion  of  a  Body.       329 

„  dl   dm        ,  dn  ,       . 

Substituting  the  values  of  — ,   — ,  and  — ,  given  by  (7), 


we  obtain   . 
at 


df    df  df 

dV 


du  \         fdv  \         fdw  \ 

"     >2    +    Vl*>\.    ). 


--  / 1  —  -  Vw3  +  Wb)2  )  +  m\-T,  ~  w*»i  +  li(J)o  j  +  ni  —  -  ?/w. 

Let  us  now  suppose  OX  to  be  the  fixed  line,  then 

dV     du 
/  =  1      m  =  n  =  0,    and  therefore    -r—  =  —  -  £w3  +  ww2 ;  but 

dt       dt 

—  is  now  the  acceleration  of  the  point  parallel  to  one  of 
dt 

the  axes  fixed  in  the  body ;  hence  we  have,  for  the  accele- 
rations of  a  point  parallel  to  three  rectangular  axes  fixed 
in  the  body,  the  expressions 


du  dv  dw 

—  -  I'lVz  +  t€Uf2,      -T.  ~  U'toi  +  «W:i,     -TT 

dt  dt  at 


where  it,  v,  iv  are  the  velocities  of  the  point  parallel  to  the 
axes  fixed  in  the  body. 

258.  Complete  Determination  of  the  Motion  of  a 
Body. — Every  motion  which  a  rigid  body  can  take  is  re- 
ducible to  a  motion  of  translation  and  a  motion  of  rotation. 
In  order  then  to  determine  the  motion  of  the  body,  a  point 
in  it  is  selected  (usually  the  centre  of  inertia),  and  the  motion 
of  the  body  is  reduced  to  the  motion  of  this  point,  together 
with  the  rotatory  motion  of  the  body  round  it. 

Geometrically  the  motion  may  be  represented  by  the 
rolling  of  a  cone,  fixed  in  the  body,  on  a  cone  unattached 
to  the  body,  except  at  one  point  (the  common  vertex  of 
the  cones),  the  latter  cone  undergoing  a  motion  of  trans- 
lation. If  the  two  cones  and  the  rate  at  which  the  one 
rolls  on  the  other  are  known,  as  well  as  the  position  in  the 
body  of  their  common  vertex,  its  velocity  at  each  instant, 
and  the  path  which  it  describes,  then  the  motion  of  the  body 
is  completely  determined. 


330  Kinematics  of  a  Rigid  Body. 

It  is  usually  most  convenient  to  consider  the  motion  of 
translation  and  the  motion  of  rotation  separately.  The  in- 
vestigation of  the  former  motion  is,  as  we  have  seen  (Art. 
205),  reducible  to  the  problem  of  the  motion  of  a  particle. 
The  latter  motion  is  completely  determined  if  we  can  assign 
at  each  instant  the  position  of  the  body  and  its  velocities  of 
rotation  in  reference  to  axes,  through  the  centre  of  inertia, 
whose  directions  are  fixed  in  space. 

The  equations  of  Kinetics  usually  give  the  velocities  of 
rotation  round  axes  fixed  in  the  body ;  but  in  order  fully 
to  determine  the  motion,  it  is  necessary  to  ascertain  the 
effect  of  these  velocities  when  the  position  of  the  body  is 
referred  to  axes  whose  directions  are  fixed  in  space.  As  the 
points  of  intersection  of  these  two  sets  of  axes  coincide,  the 
velocities  of  rotation  have  no  effect  on  the  motion  of  this 
point  0 ;  and  therefore,  so  far  as  the  angular  velocities  are 
concerned,  we  may  regard  0  as  fixed,  not  only  in  the  body, 
but  also  in  space. 

Call  the  space-axes  OX,  OY,  OZ;  the  body-axes  OA, 
OB,  OC,  each  set  being  rectangular. 

Round  the  point  0  as  centre  describe  a  sphere,  and  let 
the  axes  meet  it  at  the  points  X,  Y,  Z,  A,  B,  C. 

Three  independent  angles  are  required  to  determine  the 
position  of  the  body  in  space. 

Those  which  are  probably  the  best  adapted  for  the  solu- 
tion of  the  problem  are  the  angular  coordinates  of  the  point 
C,  or  of  the  line  OC,  and  the  angle  0,  which  the  plane  CO  A 
makes  with  the  plane  ZOC.  It  is  obvious  that  the  position 
of  OC  fixes  the  plane  A  OB,  but  does  not  determine  the 
position  of  the  lines  OA  and  OB  in  this  plane.  Hence, 
when  C  is  fixed,  if  the  angle  $  which  the  plane  CO  A  makes 
with  the  plane  ZOC  be  given,  the  position  of  the  body  is 
completely  determined.  The  angular  coordinates  of  OC  are 
0,  the  angle  which  it  makes  with  OZ,  and  \p,  the  angle 
which  the  plane  COZ  makes  with  the  plane  XOZ. 

Suppose  now  that  the  body  has  three  velocities  of  rota- 


Complete  Determination  of  the  Motion  of  a  Body.       331 

tion  :  wx,  round  OA  ;    w2  round  01? ;    and  w3  round  OC,  in 
the  direction  of  the   arrow   heads.      We   have   to   express 

— ,  ^,  and  ^  in  terms  of  these  velocities,  remembering  that 

dt    dt  dt 

the  changes  of  0,  <£,  and  \jj  are  caused  solely  by  w„  w2,  w3. 


The  motion  of  the  point  C  on  the  sphere  is  unaffected  by  w3. 
If  the  radius  of  the  sphere  be  unity,  the  point  C  has  two 
velocities,  wi  and  w2,  along  the  tangents  to  the  great  circles 
BC  and  CA.  Eesolving  these  velocities  along  the  great 
circle  ZC,  and  at  right  angles  to  it,  we  have 


dO 
dt 


o)->  cos  <p  +  (jji  sin  0, 


sin  6  —  =  w2 
dt 


sm<£  -  wi  cos  0. 


(10) 


(11) 


These  equations  are  obvious,  since  the  arc  of  a  small  circle, 
on  a  sphere  whose  radius  is  unity,  is  equal  to  the  angle  sub- 
tended at  its  pole,  multiplied  by  the  sine  of  the  spherical 
radius.  As  regards  the  algebraical  signs  it  is  well  to  ob- 
serve that  if/  is  counted  from  ZX  towards  ZY  ;  and  that  <p  is 


332 


Kinematics  of  a  Rigid  Body. 


positive  and  acute,  when  E  lies  in  the  quadrant  AB. 

The  value  of  ~  is  easily  obtained   by   considering  the 

motion  of  the  point  A  in  the  plane  AB.  The  angular 
velocity  of  A  in  the  plane  AB  is  w3,  but  it  is  also  the  com- 
ponent along  the  great  circle  AB  of  the  velocity  of  S  together 
with  the  rate  of  increase  of  SA,  i.e.  of  </> ;  and  since  SZ  and 
8 A  are  at  right  angles,  and  S  lies  on  ZC  at  a  distance  90°  -f  0 

from  Z,  the  velocity  of  S  along  AB  is  cos  Q-tt,  therefore 


nd\L      dd>       ,  d(f> 

6j3  =  cos  0  -jj  +  -77,  whence  -~ 

dt      dt  dt 


(*)* 


dt 


The  angles  made  use  of  by  Laplace  in  his  solution  of  the 
problem  of  Precession  and  Nutation  are  somewhat  different 
from  those  considered  above.  Laplace  supposes  that  a  point 
which  is  moving  from  X  to  Y  approaches  nearer  to  C  after 
passing  E,  and  he  further  places  E  behind  A  and  X.  In 
this  way  the  various  lines  and  planes  assume  the  positions 
represented  in  the  accompanying  diagram. 


The  angles  employed  by  Laplace  are  ZOC,  which  we  may 
denote  by  0',  EOA,  or  $',  and  XOE,  or  ;//.  the  last  being 
positive  when  E  is  behind  X.  Taking  into  account  the  mode 
in  which  Laplace  supposes  the  axes  to  be  situated,  we  have, 


Screws  and  Twists.  333 

then,  0'  =-  0,  $'  =  <£  -  |tt,  ^'=  Jr  -  if/,  and  equations  (10), 
(11),  and  (12)  become,  "by  substitution, 

—  =  w2  sin  0  -  oil  COS  0 
sin  0'  — -  =  wi  sin  <£'  +  ai3  cos  cj>'     y  •  (13) 

-£-  =  ai3  +  COS  0  -~ 

dt  dt  j 

259.  Screws  and  Twists. — It  was  shown  in  Article 
252  that  a  body  can  be  moved  from  any  one  position  into 
any  other,  by  a  translation  combined  with  a  rotation,  round 
an  axis  through  any  arbitrary  point  of  the  body. 

The  translation  may  be  resolved  into  two — one  parallel 
to  the  axis  of  rotation,  and  the  other  at  right  angles  thereto. 
The  latter  translation,  along  with  the  rotation,  may  be  re- 
placed by  a  pure  rotation  round  a  parallel  axis,  and  so  the 
whole  motion  will  consist  of  a  translation  parallel  to  a  certain 
fixed  line  and  of  a  rotation  round  it.  Such  a  motion  is  simi- 
lar to  that  of  a  nut  on  a  screw,  and  is  called  a  Twist.  Hence 
a  body  can  be  moved  from  any  one  position  into  any  other  by 
means  of  a  twist. 

In  order  to  determine  a  screw  it  is  necessary  to  specify — 
(1)  the  position  and  direction  of  the  line  round  which  the 
rotation  is  effected,  or  the  cans  of  the  screw ;  and  (2)  the  ratio 
of  the  translation  to  the  rotation.  This  last  is  a  linear 
magnitude,  and  is  called  the  pitch  of  the  screw.  In  order  to 
determine  a  twist,  we  must,  in  addition  to  the  screw  round 
which  it  is  effected,  specify  its  amplitude,  i.  e.  the  magnitude 
of  the  rotation. 

The  twist  by  which  a  body  can  be  moved  from  any  one  position 
into  any  other  is  in  general  unique. 

This  readily  appears  from  considering  that  if  two  posi- 
tions of  a  body  are  given,  the  magnitude  of  the  correspond- 
ing rotation  and  the  direction  of  its  axis  are  invariable ;  and 
that  if  two  positions  of  a  plane  figure  in  its  own  plane  are 


334  Kinematics  of  a  Rigid  Body. 

given,  the  position  of  the  corresponding  centre  of  rotation  is 
thereby  determined. 

The  same  thing  is  proved  directly  by  Sir  Eobert  Ball 
(to  whom  the  Theory  of  Screws  is  principally  due),  as  fol- 
lows : — 

Any  point  of  the  body,  which  lies  on  the  axis  of  the 
twist,  must  continue  thereon  after  the  motion.  If,  therefore, 
the  motion  could  be  effected  by  two  different  twists,  there 
would  be  two  different  lines  along  which  points  of  the  body 
would  continue  throughout  the  motion.  In  order  that  this 
should  be  possible,  the  lines  must  be  parallel,  and  the  motion 
one  of  pure  translation. 

If  two  successive  positions  of  a  body  in  motion  are  infi- 
nitely near  each  other,  the  twist  by  which  it  can  be  brought  from 
the  one  position  to  the  other  is  the  actual  motion  of  the  body. 
We  see  then  that  the  most  general  motion  of  a  rigid  body 
consists  of  a  succession  of  twists.  The  screw  round  which  it 
is  twisting  at  any  instant  is  called  the  instantaneous  screw.  As 
the  position  of  a  straight  line  in  space  is  determined  by  four 
independent  quantities,  five  magnitudes  must  be  assigned  to 
determine  a  screw.  In  order  to  determine  a  twist,  its  ampli- 
tude, and  the  pitch,  as  well  as  the  position  of  the  axis,  of  the 
corresponding  screw,  are  required.  Hence  the  motion  of  a 
rio-id  body  in  general  depends  on  six  independent  variables, 
and  we  see,  as  in  Article  215,  that  a  rigid  body  entirely  un- 
restrained has  six  degrees  of  freedom. 

260.  Composition  of  Twists. — If  a  body  receive  in 
succession  two  twists  whose  amplitudes  are  infinitely  small, 
the  order  in  which  they  are  effected  is  indifferent,  and  the 
resulting  change  of  position  may  be  produced  by  a  single 
twist,  which  is  the  resultant  of  the  two  former. 

More  symmetrical  results  are  obtained,  if  instead  of  seek- 
ing for  the  twist  which  is  the  resultant  of  two  others,  we 
inquire  how  three  twists  having  infinitely  small  amplitudes 
must  be  related,  in  order  that  the  position  of  a  body,  after 
being  affected  by  them,  may  remain  unaltered. 

The  question  proposed  may  be  solved  directly,  but  the 
method  of  solution  devised  by  Sir  Eobert  Ball  leads  to  results 
of  a  more  instructive  character.  This  mode  of  solution  will 
be  found  in  Example  14. 


Examples.  335 


Examples. 

1.  Determine  the  velocity  with  which  the  plane  of  the  horizon,  at  a  place 
whose  latitude  is  given,  turns  round  a  vertical  axis. 

Ans.  w  sin  A,  where  a>  is  the  earth's  angular  velocity,  and  A  the  latitude. 

2.  If  the  velocities  of  rotation  of  a  body  round  three  rectangular  axes  are 
given  in  terms  of  the  time,  show  how  to  determine— (1)  the  velocity  of  rotation 
round  the  instantaneous  axis  ;  (2)  the  position  of  the  instantaneous  axis  ;  (3)  the 
equation  of  the  cone  which  is  the  locus  of  the  instantaneous  axis. 

3.  If  the  velocities  of  rotation  round  three  rectangular  axes  are  proportional 
to  the  time  which  has  elapsed  from  a  given  epoch,  the  position  of  the  instantaneous 
axis  is  fixed. 

4.  If  the  accelerations  of  rotation  round  three  rectangular  axes  are  constant, 
the  instantaneous  axis  lies  in  a  fixed  plane. 

5.  If  6,  <p,  \p,  m,  a>2,  »3  have  the  same  significations  as  in  Art.  258,  show 
that 

de        .    „  d* 

Q}\  =  smd> sin  0  cos  d>  ~. 

Y  dt  r  dt  ' 

de  d\b 

o>2  =  cos  <p  —  +  sin  6  sin  <j>  -L, 

dd>  d\L 

at  at 

6.  A  body  is  rotating  round  a  fixed  point  0.  If  OX,  OY,  OZ  he  rectangular 
axes  fixed  in  space,  and  OA,  OB,  OC  rectangular  axes  fixed  in  the  body ;  and 
if  the  direction  cosines  of  the  latter  referred  to  the  former  be,  respectively, 
o\,  <*2>  «3 ;  bit  i>i,  bz ;  0i,  ci,  cz ;  show  that 


dai 

—  =b\0)z  —  Ci<»2, 

dt 

dbx 

— -=Cl«l-0lft>3, 

dt 

dc\ 
at 

dai     , 

— -  =  Or,  0>3  -  C-i,  0>z, 

at 

db2 

-j~  —  C2<a\  —  a%  (oz, 

dt 

dc2 

—  =  a%  (Oi  —  02  (o  i , 

dt 

daz 

—  =  03  »3  -  ^3  ^2, 

at 

dbz 

—  =  «3«1  -«3W3, 

at 

dcz 

-J-  —  OZ  (02  —  03O>1, 

dt 

where  «i,  a>2,  (oz  are  the  angular  velocities  of  the  body  round  OA,  OB,  OC. 

7.  Deduce  equations  (10),  (11),  (12),  Art.  258,  from  equations  (7),  Art.  255. 

8.  A  body  receives  in  a  given  order  rotations  of  finite  magnitude  round 
two  axes  fixed  in  space,  or  in  the  body,  and  meeting  in  a  point.  Find  the  posi- 
tion of  the  axis,  a  single  rotation  round  which  would  bring  the  body  into  the 
same  position,  and  determine  the  magnitude  of  the  resultant  rotation. 


336 


Kinematics  of  a  Rigid  Body. 


This  question  is  solved  in  a  manner  similar  to  that  employed  in  Examples  3 
and  4,  Art.  226  ;  the  construction  in  the  present  case  being  on  the  surface  of  a 
sphere  instead  of  a  plane. 

When  the  rotations  round  the  given  axes  are  in  the  same  direction,  the 
resultant  rotation  is  double  the  supplement  of  the  vertical  angle  of  a  spherical 
triangle,  whose  base  and  base  angles  are  the  angle  between  the  axes  and  the 
semi- amplitudes  of  the  rotations  round  them. 

9.  A  rigid  body  receives  a  motion  of  translation,  whose  components,  parallel 
to  the  axes,  are  a,  b,  c,  and  a  rotation  0  round  an  axis  fixed  in  the  body,  which, 
at  the  beginning  of  the  motion,  coincides  with  the  axis  of  z.  Determine  the 
position  and  pitch  of  the  screw,  a  twist  round  which  would  bring  the  body  into 
the  same  position  ;  and  find  the  amplitude  of  the  twist. 

The  screw  passes  through  a  point  whose  coordinates  are 


a  sin  h9 


b  sin  \-d  +  a  cos  hd 


Pitch  of  screw  = 


2  sin  %d 
Amplitude  of  twist 


10.  A  body  receives,  in  succession,  rotations  of  finite  magnitude  round  two 
non-intersecting  axes  a,  b,  either  fixed  in  space  or  fixed  in  the  body  :  if  d  be  the 
shortest  distance  between  the  lines  a  and  b ;  Q  and  Q'  the  amplitudes  of  the 
rotations  round  them  ;  e  the  angle  between  them  ;  tp  the  amplitude  of  the  twist 
equivalent  to  the  motion ;  and  p  the  pitch  of  its  screw  ;   prove  that 

\p§  sin^<J>  =  d  sin^0  sin  |0'  sin  e. 

(This  theorem  is  due  to  Rodrigues  :  I^iouville,  T.  5,  p.  390.) 

Take  the  shortest  distance  between  a  and  b  for  axis  of  y  ;  the  point  of  inter- 
section of  this  line  with  b  lor  origin  0  ;  and  a  parallel  to  a  for  axis  of  z. 


After  the  body  has  received  its  rotation  round  a,  suppose  it  receives  in  suc- 
cession two  equal  and  opposite  rotations  round  OZ,  the  first  of  these  being  equal 
and  opposite  to  that  round  a.  These  rotations,  being  equal  and  opposite,  do  not 
change  the  position  of  the  body. 


Examples.  337 

First,  suppose  a  and  b  to  be  fixed  in  space,  then  so  also  is  OZ. 

The  rotation  round  a  and  the  equal  and  opposite  one  round  OZ  are  (Ex.  5, 
Art.  226)  equivalent  to  a  translation,  whose  magnitude  is  2d  sin  \9,  and  whose 
direction  lies  in  the  plane  XOY,  and  is  at  right  angles  to  a  line  OP  which 
makes  with  OY  an  angle  —  \B. 

Describe,  a  sphere  round  0  as  centre,  and  let  B  be  the  point  in  which  it  is 
met  by  b,  then  ZB  =  e.  The  axis  of  the  rotation,  which  is  equivalent  (Ex.  8) 
to  the  rotations  round  OZ  and  b,  meets  the  sphere  in  B,  and  the  direction  of 
translation  meets  it  in  T ;  where 

TX  =  l9,     BZX=\9,     ZBB  =  \9'. 
Then,  by  Ex.  8, 

TBB  =  \<p, 

and  we  havej?^  =  component  of  translation  parallel  to  axis  of  screw 

=  2d  sin  \9  cos  TB  =  2d  sin  |0  sin  ZB, 
whence 

\p<$>  sin  |0  =  d  sin  |0  sin  ZB  sin  \ <p  =  d  sin  \9  sin  |0'  sin  e. 

Secondly,  if  «  and  b  be  fixed  in  the  body,  so,  likewise,  is  OZ.  In  this  case 
the  line  OP  becomes  OP' ,  which  makes  an  angle  ^9  with  OY,  and  the  points 
I7  and  B  become  T  and  B! ,  where  XT'  =  ±9,  B'BZ  =  %6'.  Then  (by  Ex.  8) 
BB'  T  =  |  <£,  and  the  result  is  obtained  in  the  same  manner  as  before. 

11.  A  body  receives  twists,  having  infinitely  small  amplitudes,  round  two 
screws  intersecting  at  right  angles.  Determine  the  amplitude  of  the  resultant 
twist,  and  the  position  and  pitch  of  its  screw. 

Take,  for  the  axes  of  x  and  y,  the  axes  of  the  screws ;  let  their  pitches  be 
p  and  q,  and  the  amplitudes  of  the  twists  round  them  9  and  <p. 

The  rotations  9  and  <p  are  equivalent  to  a  single  rotation  \p  round  an  axis 
lying  in  the  plane  cey,  and  making  an  angle  A  with  the  axis  of  x,  where 

\f/  cos  A  =3  9,     \p  sin  A  =  (p. 

The  translations  p9  and  q<p  are  equivalent  to  a  translation 

p9  cos  A  +  q<p  sin  A  =  (p  cos2  A  +  q  sin2  A)  \p 
along  the  axis  of  the  resultant  rotation,  and  to  a  translation 

q<p  cos  A  -  p9  sin  A  =  (q  —  p)  ^  sin  A  cos  A 

at  right  angles  thereto.  The  latter  translation,  together  with  the  rotation  ip 
round  the  axis  through  the  origin,  is  equivalent  to  a  rotation  ty  round  a  parallel 
axis  passing  through  a  point  on  the  axis  of  z,  whose  distance  from  the  origin  is 

(q  —  p)  sin  A  cos  A. 

Hence  the  position  of  the  axis  of  the  screw  corresponding  to  the  resultant  twist 
is  given  by  the  equations 

y  =  x  tan  A,     z  =  (q  -  p)  sin  A  cos  A, 
Z 


338  Kinematics  of  a  Rigid  Body. 

<b 
where  tan  A  =  — : 

6 

and  its  pitch  by  the  equation    r  =  p  cos2  A  +  q  sin2  A. 

Also,  if  \p  be  the  amplitude  of  the  resultant  twist,  we  have 


12.  Any  screw,  a  twist  round  which  is  the  resultant  of  twists  round  two 
given  screws  intersecting  at  right  angles,  lies  on  a  surface  determined  by  the 
given  screws ;  and  its  pitch  depends  on  the  angle  wbich  its  direction  makes 
with  one  of  these  screws.  The  amplitudes  of  the  twists  are  supposed  to  be 
infinitely  small. 

If  Ave  eliminate  A  from  the  equations  of  the  last  example,  which  define  the 
position  of  the  screw  belonging  to  the  resultant  twist,  we  obtain 

z  (a?  +  2/2)  -  {q  -  p)  xy  =  0, 

which  is  the  equation  of  the  surface. 

This  surface  is  called  the  cylindroid  by  Sir  Eobert  Ball. 

The  pitch  r  of  the  screw  is,  as  shown  in  the  last  example,  given  by  the 
equation 

r  =  p  cos2  A  +  q  sin2  A. 

If  we  describe  in  the  plane  of  xy  the  conic  whose  equation  is 

px2  +  qy2  =  e3, 

where  e  is  a  linear  constant,  the  square  of  the  reciprocal  of  any  diameter  of  this 
conic  is  proportional  to  the  pitch  of  the  parallel  screw  on  the  cylindroid.  When 
a  screw  is  spoken  of  as  belonging  to  the  cylindroid,  it  is  understood  that  not 
only  is  its  axis  one  of  the  generating  lines  of  the  surface,  but  also  that  its  pitch 
is  defined  in  the  manner  just  mentioned. 

13.  Prove  that  any  two  screws  belong  to  the  same  cylindroid.  Also  two 
screws  being  given,  determine  the  cylindroid  to  which  they  belong. 

Take  the  common  perpendicular  to  the  two  screws  as  axis  of  z  ;  we  have  then 
to  determine  the  position  of  the  origin  and  of  the  axis  of  x,  and  the  magnitudes 
of  the  quantities  p  and  q,  so  as  to  satisfy  the  equations 

r\  =  p  cos2  Ai  +  q  sin2  Ai,         z\  =  (q  —  p)  sin  Ai  cos  Ai, 

r2  =  p  cos2  A2  +  q  sin2  A2,        «2  =  (q  —  p)  sin  A2  cos  A2, 

A  =  Ai  —  A2,  h  =  z\  —  Z2, 

where  A  is  the  angle,  and  h  the  distance  between  the  given  screws.  As  the 
number  of  quantities  at  our  disposal  is  equal  to  the  number  of  equations  to  be 
satisfied,  it  is  always  possible  to  determine  a  cylindroid  containing  the  given 
screws.     The  equations  are  solved  as  follows  : — Subtracting, 

n-  r2  =  (q  -  p)  (sin2  Ai  -  sin2  A2)  =  [q  -  p)  sin  (Ai  +  A2)  sin  (Ai  -  A2), 
h  =  (q  -  p)  cos  (Ai  +  A2)  sin  (Ai  -  A2). 


Examples,  339 

Adding, 

r\  +  r2  =  1p  -  (p  -  q)  (sin2  Ai  +  sin2  A2) 

-p  +  g  +  $  {p  -  q)  (cos  2\i  +  cos  2a2) 
-P  +  2  +  (P  ~  <?)  cos  (Ai  +  A2)  cos  (Ai  -  A2), 
*i  +  -2  =  (j  -j?)  sin  (Ai  +  A2)  cos  (Ai  -  A2) ; 

hence  (q  -  p)  sin  (Ai  +  A2)  =  A — -2,     zx  +  z2  =  (n  -  r2)  cot  A, 

sin  ^1 

(?  -  p)  cos  (Ai  +  A2)  =  - — -,     zx  -  z2  =  h, 
sin  ud. 

P  +  q  =  n  +  n  +  A  cot  ^4,    Ai  -  a2  =  ^4, 
and  the  mode  of  completing  the  solution  is  obvious. 

14.  A  body  receives  three  twists  having  infinitely  small  amplitudes.  Deter- 
mine the  relations  between  the  twists,  in  order  that  the  position  of  the  body 
should  remain  ualtered. 

Sir  Robert  Ball's  solution  is  as  follows  : — 

Determine  the  cylindroid  containing  two  of  the  screws.  Take  its  screws 
intersecting  at  right  angles  for  axes  of  x  and  y.  Let  flx,  n2,  n3  be  the  ampli- 
tudes of  the  three  twists,  and  Ai,  A2,  A3  the  angles  which  their  screws  make 
with  the  axis  of  x.  If  the  third  screw  belongs  to  the  cylindroid  containing  the 
other  two,  and  if  the  angle  it  makes  with  the  axis  of  x  and  the  amplitude  of 
the  corresponding  twist  satisfy  the  equations 

Oi  n2  n3 


sin  (A2  -  A3)      sin  (A3  -  Ai)      sin  (Ai  -  A2)' 

the  twists  compensate  each  other. 

In  fact  each  twist  can  be  resolved  into  two  round  the  screws  lying  along  the 
axes  of  x  andy.  The  whole  motion  is  thus  reduced  to  two  twists  round  these 
screws  ;  and  if  the  amplitudes  of  these  twists  are  zero,  the  body  remains  undis- 
turbed. But  the  equations  above  are  the  conditions  that  the  rotations  round  the 
axes  of  x  and  y  should  be  zero,  and  these  rotations  are  the  amplitudes  of  the 
twists. 

As  the  twist  by  which  a  given  motion  can  be  effected  is  unique,  there  is  only 
one  twist  by  which  two  given  twists  can  be  compensated ;  and,  therefore,  if 
three  twists  compensate  each  other,  the  third  screw  must  belong  to  the  cylin- 
droid containing  the  other  two,  and  the  above  equations  must  hold  good. 

15.  A  body  is  moving  round  a  fixed  point.     Determine  the  accelerations  of 
any  point  parallel  and  at  right  angles  to  the  instantaneous  axis  of  rotation. 
Taking  three  lines  fixed  in  space  through  the  fixed  point  as  axes, 


dx  dy  dz 

<zx- 

z2 


dt  -jy'    ft"-**"-**    ^  =  «-y-^«; 


340 


Kinematics  of  a  Rigid  Body. 


,     .      .        „      dx      dy      dz 
differentiating,   and  substituting  for  — ,     -|,     — ,   from  these  equations,  we 

obtain, 

d2x                                         no           duy         da)z 
— -  =  u)x  (u>z  x  -j-  wy  y  +  w,  z)  -  (azx  +  z  — y  — 

dt-  at 


d2y 


— -  =  wy  (ux  x  +  a>y  y  +  ws  z)  -  co2  y  +  x  — 
dtz  <*t 


dt 

dcox 


<Pz 
dt* 


=  o>z  (o>x  x  +  coy  y  -f-  w3  z)  -  a>2  z  +  y 


dcox        da. 


dt 


dt  ' 


remembering  that 


.,2  = 


oo  x2  +  toy*  +  &>*2 


Let  us  now  suppose  the  axis  of  z  to  coincide  with  01,  the  instantaneous  axis, 
then  oox  =  0,  w,j  =  0,  wt  =  «.  Let  the  plane  of  xz 
pass  through  01',  the  consecutive  position  of  the 
instantaneous  axis.  Measure  off  01  proportional 
to  co  on  OZ,  and  take  OF  proportional  to  the  cor- 
responding angular  velocity  u  +  doo  ;  draw  IP 
perpendicular  to  01 ;  then  w  +  do*  round  01'  is 
resolvable  into  OP  round  OZ,  and  PP  round  OX. 

Let  JT'OP  =  dty  ;  then  PP  =  OP  dty ;  therefore 


dbOx 

~dt 


'dt 


=  w^, 


if  the  angular  velocity  of  the  instantaneous  axis  be  denoted  by  fy.     Also 


£-•* 


— ,     since    d«>zoz  IP  =  OF  -  01. 

dt 


Introducing  tbese,  we  obtain 


d*x 

dt* 


duo 
• X 

dt 


d?-y      doo  ,  .        d2z         . 


16.  Find  the  position  of  the  acceleration-centre  in  a  body  rotating  round  a 
fixed  point. 

The  only  acceleration-centre  which  in  general  exists  is  [the  fixed  point 
itself. 

17.  A  body  is  moving  round  a  fixed  point  0.  If  perpendiculars,  whose  lengths 
are  p  and  q,  be  let  fall  from  any  point  A  of  the  body  on  01,  the  instantaneous 
axis  of  rotation,  and  on  OJ,  that  of  angular  acceleration  ;  prove  that  the  total 
acceleration  of  A  is  the  resultant  of  two  components,  oo'zp  along  p  and  tig  per- 
pendicular to  the  plane  AOJ,  where  a>  and  o-  are  the  resultant  angular  velocity 
and  angular  acceleration  of  the  body. 


Examples.  341 

If  x,  y,  z  be  the  coordinates  of  A  referred  to  space  axes  through  0,  and  r 
be  the  distance  OA,  the  equation  for  the  acceleration  x  may  be  written  (Ex.  15), 


x 
where 


/     tox  toy  W«\  )  ( 'toy  z         a>z  y\ 

[x  —  +  y  -  +z-  )-x    -for -    , 

\     co  co  co/         J*  \  cr  r        a  r  / 


a2  =  ux   4  toy"  +  «i>s  • 

If  P  be  the  point  in  which  the  perpendicular  from  A  meets  01,  and  if 
we  consider  the  projection  of  the  triangle  OPA  on  the  axis  of  x,  we  have 
projection  of  AP  =  projection  of  OP  -  projection  of  OA.  From  this  it  is  plain 
that  the  term  by  which  co-  is  multiplied  in  x  is  the  projection  of  p  on  the  axis 
of  x.  Again,  if  A,  fi,  v  be  the  direction  cosines  of  the  normal  to  the  plane  AOJ, 
and  9  the  angle  between  OJ  and  OA,  we  have 

(b„  z       to%  y 

Xsm8=— ,    and    r  sin  9  =  q  ; 

a  r       ff  r 

whence  it  appears  that  the  term  by  which  <r  is  multiplied  in  x  is  q\,  or  the  pro- 
jection of  q  on  the  axis  of  x.  The  truth  of  the  theorem  above  is  now  obvious. 
This  theorem  is  due  to  Professor  Minchin. 

18.  A  body  is  rotating  round  a  fixed  point :  find  the  locus  of  a  point  whose 
acceleration  along  its  path  at  any  given  instant  is  zero. 

As  the  path  at  the  instant  touches  a  circle,  having  its  centre  on  the  instan- 
taneous axis  and  its  plane  at  right  angles  thereto,  if  jj  be  the  distance  of  any 
point  from  the  axis, 

,  ....  y  d2x      x  d2i/      x2  +  yz  du>        .  xz 

the  tangential  acceleration  = — r  H —  = u\p  — . 

p  dt2      p  dt2  p       dt  p 

The  required  locus  is  therefore  the  cone 

dco 

—  (x2  +  y2)  -  wipxz  =  0. 
dt 

19.  Show  that  a  point  whose  normal  acceleration  at  right  angles  to  the 
instantaneous  axis  vanishes  lies  on  the  cone 

to  {x2  +  y2)  +  fyz  =  0. 

20.  A  body  is  rotating  round  a  fixed  point :  determine  at  any  instant  the 
positions  of  the  osculating  plane,  and  of  the  principal  normal,  to  the  path 
described  by  one  of  its  points. 

The  normal  plane  to  the  path  is  the  plane  passing  through  the  point  and  the 
instantaneous  axis.  Hence  the  perpendicular  to  the  osculating  plane  is  the 
intersection  of  this  plane  with  its  consecutive  position.  Again,  the  direction  of 
the  principal  normal  coincides  with  that  of  the  resultant  normal  acceleration  ; 
hence,  if  v  be  the  angle  the  principal  normal  makes  with  the  instantaneous  axis, 

co;;2+  &jz 
tan  v  = : . 

to 


342 


Kinematics  of  a  Rigid  Body. 


21.  Find  the  radius  of  curvature  of  the  path  of  any  point  of  the  body. 
If  iVbe  whole  normal  acceleration, 


P  = 


upz 


N 


V{(«p3  +  ^)8+^yV} 


22.  A  body  is  moving  in  any  manner.  Determine  the  accelerations  of  a 
point  parallel  and  at  right  angles  to  the  axis  of  the  instantaneous  screw. 

Let  xo,  l/o,  zq  be  the  coordinates  of  a  point  fixed  in  the  body,  {,  77,  £the  co- 
ordinates of  any  point  referred  to  x0,  y0,  z0  as  origin  ;  then 


d2x 
dP 


d2x0 


da* 


do 


— — -  +(Dx(wx£+  wyTj-f  uzQ  -co2! -f  C~r. -  -V-Ji 


dt2 


dt 


dt 


**y         ^Vo,  f       t.  .  fy 

di?  =  It2"     WyV**l  +  a,J/7?  +Wz&~  u~ 


dojs 


-c 


do}z 

~dTi 


d~zd2z0  dcox 


dt 


Take  as  xo,  1/0,  zo  that  point  of  the  body  which  at  the  instant  coincides  with 
the  point  0  on  the  instantaneous  screw  in  space  which  is  nearest  the  consecu- 
tive position  of  the  instantaneous  screw.  If  C  be  the  ruled  surface  in  space 
generated  by  the  positions  of  the  instantaneous  screw-axis,  0  will  be  the  point 
of  intersection  of  the  instantaneous  screw- axis  with  the  line  of  striction  on  C. 
Let  00'  be  an  element  of  this  line  of  striction. 
At  the  time  t  +  dt  the  body  is  twisting  round  a 
screw  through  0'.  Let  Tbe  the  velocity  of  trans- 
lation at  the  time  t,  and  T  and  w'  the  velocities 
of  translation  and  rotation  at  the  time  t  4-  dt. 

Now,  the  velocity  of  rotation  00'  round  O'S 
(screw-axis  through  0')  is  equivalent  to  o»' round 
01'  (parallel  to  O'S),  and  a  velocity  of  trans- 
lation oj'.OO'  at  right  angles  to  00'  and  01'. 

The  velocity  of  rotation  w'  round  01'  is 
equivalent  to  u>  round  OZ,  andwWi//  round  OX. 
Hence,  at  the  time  t  +  dt,  the  point  Xoyozo 
has  two  velocities  of  translation :  T'  along  OZ, 
and  (oj'.OO'  +  T'd$)  along  OX.  Again,  as  00' 
is  infinitely  small  of  the  first  order,  the  velocity 
of  translation  along  OZ  resulting  from  00'.  (a' 
is  infinitely  small  of  the  second  order.  At  the  time  t  the  point  x0ijqZo  had  the 
velocity  T  along  OZ.  Hence,  if  U  be  the  velocity  of  translation,  and  ^  the 
angular  velocity  of  the  axis  of  the  instantaneous  screw,  at  the  instant,  we  have 


=  *, 


d2x0 
dt* 


=  .U+T*.        ^=0, 


d2z0 

~~d~F 


dT 
dt' 


Examples.  343 

AlSO  (I)X  =0,  (l)y  ■ 

whence 


dccx        '      dcov 
=  0,     coz=v,      —  =co*,     —  =  0, 

dwz 

~dt  ' 

dw 

~~  It 

dH         _      _.         „,.      <fw 

5T -•*+*»- *«-;** 

d~z     dT       . 
df~  ~  dt  +  wlhm 

du)x        . 

duly 

duiz      duj 

dt         r 

— "  =  0, 

dt 

dt        dt 

23.  A  body  is  moving  in  any  way  :  determine  the  position  of  the  accelera- 
tion-centre at  any  instant. 

Its  coordinates  are  formed  from  the  equations  of  the  last  example,  by  making 

df-         '       dt2         '       dt2 

24.  A  body  is  moving  in  any  way  ;  the  acceleration  at  any  instant  of  any 
point  is  the  same  as  if  the  body  were  rotating  round  the  acceleration-centre  as 
an  absolutely  fixed  point. 

Express  the  accelerations,  by  the  last  example,  in  terms  of  the  coordinates 
relative  to  the  acceleration-centre  as  origin,  and  the  results  are  the  same  as  the 
expressions  of  Ex.  15  would  become,  if  we  made 

(Dx=   0,         QJy=0,        (J)z  — 

The  theorem  is  likewise  obvious,  a  prori. 

The  theorems  contained  in  the  Examples  given  above  are  taken  from  Schell's 
Theorie  der  Bewegung  und  der  Erafte,  to  which  the  student  is  referred  for  more 
extended  investigations  on  the  subject. 

25.  Show  that  the  theorem  of  Ex.  17  holds  good  for  a  body  moving  freely 
provided  the  acceleration- centre  be  substituted  for  the  fixed  point  O. 

This  extension  of  Ex.  17  is  due  to  Professor  Minchin. 

26.  A  right  circular  cone  is  rolling  on  another  fixed  in  space,  the  two  cones 
having  a  common  vertex.  Given  the  velocity  of  rotation  of  the  rolling  cone, 
determine  the  velocity  with  which  the  plane  passing  through  the  instantaneous 
axis  turns  round  the  axis  of  the  fixed  cone. 

The  normal  plane  through  the  instantaneous  axis  contains  the  axes  of  both 
cones.  Hence  the  angle  between  the  two  axes  remains  invariable  ;  and  a  point 
on  the  axis  of  the  rolling  cone  describes  a  circle  having  its  centre  on  the  axis  of 
the  fixed  cone,  and  its  plane  perpendicular  thereto.  It  is  also  at  any  instant 
rotating  round  the  instantaneous  axis.  If  we  equate  the  two  expressions  for  the 
velocity  of  this  point,  we  get 

a  sin  r  =  a  sin(C+  r),  (1) 

or  -  w  sinr  =  nsin(C-r),  (2) 

or  sin  r  =  nsin(r- C),  (3) 


344 


Kinematics  of  a  Rigid  Body. 


where  w  is  the  angular  velocity  of  the  rolling  cone  round  the  instantaneous  axis ; 
Cl  the  angular  velocity  of  the  plane  containing  the  instantaneous  axis  round  the 
axis  of  the  fixed  cone  ;   C  and  r  the  semi-angles  of  the  fixed  and  moving  cones. 


The  first,  second,  or  third  formula  is  to  he  used,  according  as — (1)  the  cones  are 
outside  one  another,  having  convex  surfaces  in  contact ;  (2)  the  rolling  cone  is  a 
small  cone  rolling  inside  a  larger  one  ;  (3)  the  rolling  cone  is  the  larger  cone, 
and  rolls  outside  a  smaller  fixed  cone,  which  it  contains  within  it. 

In  each  of  these  figures  00  is  the  axis  of  the  fixed  cone ;  Or  of  the  rolling 
cone  ;  and  01  the  instantaneous  axis.  If  the  angles  are  supposed  to  contain 
their  signs  implicitly,  each  heing  measured  from  01,  the  last  formula  contains 
the  other  two. 

27.  A  hody  is  moving  round  a  fixed  point.  The  motion  of  the  instantaneous 
axis  in  the  hody  heing  completely  given,  determine  its  motion  in  space. 

Descrihe  a  sphere  of  radius  around  the  fixed  point :  the  cone  C  fixed  in  space 
and  the  cone  r  fixed  in  the  body  trace  out  curves  on  this  sphere,  and  the  motion 
is  accomplished  by  the  one  curve  rolling  on  the  other.  The  osculating  circle  of 
each  of  these  curves,  as  it  passes  through  three  points  on  the  surface  of  the  sphere, 
will  be  a  circle  of  the  sphere ;  and  the  roiling  at  any  instant  will  be  the  same  as 
if  one  of  these  circles  rolled  on  the  other,  or  as  if  the  right  cone  on  the  osculating 
circle  of  r  as  base  rolled  on  the  right  cone,  having  the  osculating  circle  of  C  as 
base.     Let  r  be  the  radius  of  curvature  of  the  curve  C;  p  of  the  curve  T;  then 

sin  C  =  -,        sin  r  =  -,     and  therefore  (Ex.  26), 
a  a 


P  •       (     •        !      P 

-  =  Qsin  sin'1  - 
a  (  a 

ds 


-;} 


Now,  if  «  be  the  arc  of  the  curve  C,  and  — ■  the  velocity  of  the  point  of  contact 
of  r  along  it, 


1  ds      1  da 
r  Jt=  r  dt' 


-iP 


n  = 

where  <r  is  the  arc  of  r  ;  whence 

p      1  da- 
«-  =  --—  sin  <  sin' 
a      r  dt  {  a 

From  this  equation  r  can  be  determined  in  terms  of  t  (in  terms  of  which  p,  a>, 

and  —  are  supposed  to  be  expressed) ;  and,  as  —  =  — ,  by  eliminating  t  an 

dt  dt       at 

equation  is  obtained  between  r  and  s,  which  is  the  equation  of  the  curve  C; 
therefore,  &c. 


Kinetics  of  a  Rigid  Body.  345 


Section  II. — Kinetics. 

261.  Moments  of  Momentum  of  a  Body  having  a 
Fixed  Point. — If  x,  y,  z  be  the  coordinates  of  any  point  of 
the  body,  referred  to  space  axes  intersecting  at  the  fixed 
point  0  ;  and  Hx,  Hy,  Hz,  the  moments  of  momentum  round 
these  axes,  we  have  Mx  =  Sm  (yz  -  zy) .  Substituting  for 
y  and  z  their  values  given  by  (4),  Art.  255,  we  obtain 

Hx  =  o)xS(!/2  +  s2)  dm  -  wylxydm  -  wz\xzdm. 

Hence,  if  a,  b,  c,  i,  J,  k  be  the  moments  and  products  of 
inertia  of  the  body  at  any  instant,  round  the  space  axes, 
we  have 


sx 

= 

au)T  — 

JCWy 

-J(Oz 

By 

= 

—  k(Ox  + 

bWy 

—  iu)z 

bz 

= 

-.M  ~ 

iu)v 

+  Cd)z 

(1) 


If  the  space  axes  coincide  with  the  instantaneous  posi- 
tion of  the  principal  axes  of  the  body  at  0,  equations  (1) 
become 

B,  =  Am,  B2  =  JBuj2,  B,  =  Cw3,  (2) 

where  Bly  B2,  B3,  wi,  w2,  w3  are  the  moments  of  momentum 
and  the  angular  velocities  round  the  principal  axes  at  the 
instant ;  and  A,  B,  C  are  the  principal  moments  of  inertia 
of  the  body  for  the  point  0. 

The  resultant  moment  of  momentum  B  is  given  by  the 
equation 

B2  =  A2^2  +  B2w22  +  C2^.  (3) 

The  direction  cosines  of  the  momentum  axis  relative  to 
the  principal  axes  through  0  are  proportional  to  Awi, 
Bu)2,  Cw3. 


346  Kinetics  of  a  Rigid  Body. 

If  #,,  a2,  a3,  bh  b2,  bs,  ci9  c2,  cs  be  the  direction  cosines  of 
the  principal  axes  at  0  referred  to  the  space  axes,  we  have 

Hx  =  Awxcti  +  Bixjobi  +  CojzCx    \ 

Hy  =  Awxa2  +  Bu2b2  +  Cu)2c2    [•  (4) 

Hz  =  Awictz  +Bu)2b3  +  CwzCz    ) 

If  0  be  a  definite  point  of  the  body  not  fixed  in  space, 
equations  (1),  (2),  (3),  (4)  still  hold  good  for  the  motion 
relative  to  0 ;  the  axes  x,  y,  and  z  being  parallel  to  fixed 
directions  in  space. 

262.  Motion  of  a  Body  having  a  Fixed  Point 
under  the  Action  of  Impulses. — If  a  body  having  a 
fixed  point  0  be  acted  on  by  any  set  of  impulses,  whose 
moments  round  the  principal  axes  through  0  are  L,  M,  N; 
these  moments  are  equal  respectively  to  the  changes  in  the 
moments  of  momentum  of  the  body.     Hence  ( (2),  Art.  262), 

A  (o>,  -  W)  =  L,  B (012 -  w2)  =  M,  C (to* -  w/)  =  N,    (5) 

where  &)/,  w2,  to*',  and  u)h  to>2,  to>3  are  the"^  angular  velocities 
round  the  principal  axes  immediately  before  and  imme- 
diately after  the  action  of  the  impulses. 

In  some  cases  it  may  be  convenient  to  use  the  expres- 
sions for  Hx,  Hy,  H~  given  in  (1),  Art.  261,  and  the  moments 
GXi  Gy,  Gz  of  the  impulses  round  the  space  axes.  We  have, 
then, 

CI  (toX  -  to)/)    -   k  (b)y  -   to)/)    -  j  (to)z  -  to)/)    =     Gx      \ 
-  JC  (to)*  -  to)/)   +    b  (toly  -   to)/)   -   i  (to)a  -  to)/)    =     Gy      > .  (6) 

-j  (to)*  -  to)/)  -  »  (to)*,  -  to)/)  +  C  (to)z  -  to)/)  =   Gz     I 

263.  Vis  Viva  of  a  Body  having  a  Fixed  Point. — 

As  the  body  has  a  fixed  point,  it  is  at  any  instant  rotat- 
ing round  some  axis  through  it ;  whence  the  vis  viva  is 
7to)2,  /  being  the  moment  of  inertia  round  the  instanta- 
neous axis. 


Couple  of  Principal  Moments.  347 

Again,  since  — ,  — ,  —  are  the  direction  cosines  of  the  axis 

to        to         to 

of  rotation  referred  to  the  principal  axes  through  the  fixed 
point, 


wr2 


I  =  A  -  \+B  -3    +C  -J   ;    (Int.  Calc,  Art.  215) ; 


whence,  if  2T  or  #  be  the  w's  mwb  of  the  body,  we  have 

2T=S  =  Ato?  +  ZW  +  CtoZ\  (7) 

If  toz,  toy,  to-  be  the  velocities  of  rotation,  and  «,  6,  c, »,  &c, 
the  moments  and  products  of  inertia  of  the  body  at  any 
instant  round  space  axes  through  0,  the  general  equation  of 
the  momental  ellipsoid  referred  to  these  axes  leads  to  the 
following  expression — 

2T=  Ito2  =  citox2  +  btoy  +  Cto~r  -  2itoytoz  -  2jtoztox  -  2ktoxioy.     (8) 

264.  Couple  of  Principal  Moments. — If  a  body  be 
moving  round  a  fixed  point,  we  may  imagine  its  actual  velo- 
city at  any  instant  to  be  produced  by  an  impulsive  couple 
acting  on  it  at  the  instant.  By  the  last  Article  the  compo- 
nents of  this  couple  round  the  principal  axes  of  the  body  are 
Atoh  Btoo,  Cto3,  and  the  axis  of  the  couple  is  called  the  Axis 
of  Principal  Moments.  This  axis  coincides  at  each  instant 
with  the  momentum  axis  of  the  body  (Arts.  210,  261). 

If  a  tangent  plane  be  drawn  at  the  point  of  intersection 
of  the  instantaneous  axis  of  rotation  with  the  momental  ellip- 
soid corresponding  to  the  fixed  point  round  which  the  body 
is  rotating,  the  perpendicular  from  the  centre  on  this  tangent 
plane  is  the  Axis  of  Principal  Moments.  This  is  obvious, 
when  we  remember  that  the  direction  cosines  of  this  axis  are 
proportional  to  Atoh  Bto2,  Civ,, ;  and  those  of  the  instantaneous 
axis  of  rotation  to  wlf  w2,  w3 ;  and  that  the  equation  of  the 
momental  ellipsoid  is 

Ax2  +  Bif  +  Cz2  =  K. 

If  (j)  be  the  angle  between  the  momentum  axis  and  the 
instantaneous  axis  of  rotation,  R  the  moment  of  momentum, 


348  Kineiics  of  a  Rigid  Body. 

and  8  the  vis  viva  of  the  body,  we  have,  by  the  formula  for 
the  cosine  of  the  angle  between  two  lines  in  terms  of  their 
direction  cosines, 

Hcj  cos  ^  =  Au)x  +  Buz  +  Cwz  =  8 ; 

or 

whence  w  cos  <j>  =  -=.  (9) 

Again,  if  r  be  the  intercept  made  by  the  momental  ellip- 
soid on  the  instantaneous  axis  of  rotation,  we  have 

K         w!2  ,  „  w22     n  Wz       8 
r*  to  it)  (*)        w 

whence  r2  =  -=  w2.  (10) 

Again,  if  we  draw  a  tangent  plane  to  the  momental  ellip- 
soid at  the  point  where  it  meets  the  instantaneous  axis  of 
rotation,  the  intercept  p  made  by  this  plane  on  the  momen- 
tum axis  is  given  by  the  equation  p  =  r  cos  0,  since  the 
momentum  axis  is  perpendicular  to  the  tangent  plane. 
Hence,  if  we  substitute  for  r  and  w  cos  0  their  values  given 
by  (10)  and  (9),  we  obtain 

P  -  ^p.  (ii) 

Examples. 

1.  A  body  is  set  in  motion  by  an  impulsive  couple  whose  magnitude  is 
given  ;  find  the  direction  of  its  axis  so  that  the  initial  vis  viva  of  the  body  may 
be  a  maximum. 

The  axis  of  the  couple  must  be  the  axis  of  least  inertia  of  the  body. 

2.  A  body  having  a  fixed  point  0  is  set  in  motion  by  an  impulse,  passing 
through  a  point  P,  which  causes  P  to  move  with  a  velocity  having  a  given  mag- 
nitude and  direction  ;  determine  the  axis  of  instantaneous  rotation.         _ 

Let  the  axis  of  a;  be  a  line  through  0  in  the  direction  of  the  velocity  of  P, 
and  the  axis  of  z  the  line  OP ;  then,  if  V  be  the  given  velocity  of  P,  h  the 
distance  OP,  and  a>„,  a>9,  wz  the  angular  velocities  of  the  body  round  the  axes, 
we  have    &>*  =  0,     h  wy  —  V. 


Examples.  349 

Xow,  by  Thomson's  Theorem,  Art.  199,  the  value  of  az  must  he  such  as  to 
make  To.  minimum.     But,  Art.  263,  (8), 

dT  _ 

— —  =  —  iuy  +  ccoz.      Hence     cw-  =  iwy, 

du~ 

which  determines  wz,  and  consequently  the  axis  of  rotation. 

3.  In  Ex.  2,  when  is  the  velocity  of  rotation  around  OP  zero  ? 

Ans.  "When  OP  is  an  axis  of  the  section  of  the  momental  ellipsoid  which 
is  perpendicular  to  the  initial  motion  of  P. 

4.  In  Ex.  2,  if  the  magnitude  of  Vhe  given  as  before,  find  its  direction  so 
that  the  initial  vis  viva  of  the  body  may  be  a  maximum  or  a  minimum. 


dT  dT  dT 

By  Art.  263,  2T  =  «*  —  +  wy  —  +  wz  —  ; 

clwx  dcav  doc- 


but 


Hence 


dT 

o>x  =  0,    and  —  =  0  (Ex.  2). 

dcoz 

or    h    2     •  u  -  *2     2     B'C'  V 

11=  OCtit/~  —   IWutoz  =    Wu     = — 

c  c      h' 


where  i?'and  C  are  the  moments  of  inertia  of  the  body  round  the  axes  of  the  section 
of  the  ellipsoid  of  inertia  made  by  the  plane  yz.  The  maximum  or  minimum 
value  of  Tis  obtained,  then,  by  making  c  equal  to  C  or  to  B' ;  i.  e.  the  direction 
of  V  must  be  perpendicular  to  the  central  section  of  the  ellipsoid  having  OP  as 
an  axis. 

5.  If  a  body  be  moving  in  any  manner,  the  momentum  axis,  and  the  in- 
stantaneous axis  of  rotation  through  a  given  point  0  of  the  body,  are  the  radius 
vector  and  the  perpendicular  on  the  corresponding  tangent  plane  of  the  ellipsoid 
of  gyration  (see  Integral  Calculus,  Art.  216)  relative  to  0. 

This  is  the  reciprocal  of  the  theorem  given  in  Art.  264.  It  can  be  easily 
proved  directly  :  <t\,  o%  <tz,  and  a,  j8,  y,  being  the  angles  made  by  the  momentum 
axis  and  the  instantaneous  axis  of  rotation  with  the  principal  axes,  and  a,  b,  c 
the  principal  radii  of  gyration, 

H  cos  d\  —  A(t)\  =  ma2u  cos  a,  &c. ; 

but  if  x'y'z  be  a  point  on  the  ellipsoid  of  gyration,  a',  £',  y'  the  angles  made 
with  the  axes  by  the  perpendicular  on  the  corresponding  tangent  plane,  and^» 
the  length  of  the  perpendicular,  x'p  =  ar  cos  a',  &c. ;  therefore  if  x\  &c.  be  propor- 
tional to  cos  <r\,  &c,  a'  =  a,  &c. 

The  student  will  observe  that  H  here  denotes  the  moment  of  the  momentum 
of  the  motion  relative  to  0,  but  not  of  the  absolute  motion,  except  0  be  a  point 
fixed  in  space. 

6.  If  a  tangent  plane  to  the  ellipsoid  of  gyration  relative  to  any  point  of  a 

body  be  drawn  at  right  angles  to  the  instantaneous  axis  of  rotation  passing 

throught  the  point,  and  <p  be  the  angle  between  the  instantaneous  axis  and  the 

,.  ,  H  cosd> 

radius  vector  to  the  point  of  contact,  «  = TL. 

mp~ 

This  is  immediately  deducible  from  the  consideration  that  mp^io  =  !&  =  mo- 
ment of  momentum  round  instantaneous  axis  of  rotation  =  II  cos  <p. 


350  Kinetics  of  a  Rigid  Body. 

7.  Express  the  perpendicular  on  the  tangent  plane  to  the  ellipsoid  of  gyration 
in  terms  of  the  angular  velocity  and  relative  vis  viva. 

Ans.     p1  = -. 

mat* 

8.  Express  the  intercept  cut  off  by  the  ellipsoid  of  gyration  on  the  momentum 
axis  in  terms  of  the  relative  vis  viva  and  moment  of  momentum. 

Ans.     i22  =  — . 
mS 

9.  A  body,  having  a  fixed  point  0,  is  set  in  motion  by  an  impulse  of  given 
magnitude  and  passing  through  a  given  point  Pof  the  body;  find  the  direction 
of  the  impulse  so  that  the  initial  vis  viva  of  the  body  may  be  a  maximum,  fj 

Since  the  impulse  has  no  moment  round  the  line  OP,  the  momentum  axis 
lies  in  the  plane  perpendicular  to  OP :  also  by  Ex.  8,  the  vis  viva  is  a  maximum 
when  PL  is  a  maximum  and  R  a  minimum.  Hence  R  must  be  the  shortest  axis 
of  the  section  of  the  ellipsoid  of  gyration  made  by  the  plane  perpendicular  to 
OP,  and  the  direction  of  the  impulse  must  be  parallel  to  the  longest  axis  of  this 
section. 

10.  In  Ex.  9,  show  that  the  initial  velocity  of  P  in  the  direction  of  the  im- 
pulse is  a  maximum,  and  that  the  instantaneous  axis  of  rotation  lies  in  the  plane 
of  R  and  OP. 

Since  the  vis  viva  is  a  maximum,  so  is  the  initial  velocity  of  P  in  the  direc- 
tion of  the  impulse,  Art.  199.  Again,  if  R'  be  the  axis  major  of  the  section  of  the 
ellipsoid  of  gyration  perpendicular  to  OP,  the  plane  perpendicular  to  R'  con- 
tains the  perpendicular  on  the  tangent  plane  drawn  at  the  extremity  of  R,  that 
is,  the  plane  of  R  and  OP  contains  the  instantaneous  axis  of  rotation. 

265.  Motion  of  a  Free  Body  under  the  Action  of 
Impulses. — If  X,  Y9  Z,  be  the  components  of  any  one  of  the 

impulses,  and  u',  v,  w',  ii,  v,  w  the  components  of  the  velocity  of 
the  centre  of  inertia  G  before  and  after  the  action  of  the  im- 
pulses, the  velocity  of  G  is  determined  by  the  equations 

Wl  (u  -  u')  =  SZ,    $1  (v - %')  =  2 F,  SSR  {w - w)  =  2Z,  (12) 

where  50?  is  the  mass  of  the  body. 

Again  (Art.  209),  the  motion  of  the  body  relative  to  its 
centre  of  inertia  is  the  same  as  if  that  point  were  fixed  in 
space.  Hence  if  L,  M,  N  be  the  moments  of  the  impulses 
round  the  principal  axes  of  the  body  at  G,  and  toi,  a>2',  w3', 
wi,  w2,  w3  the  angular  velocities  of  the  body  round  these 
axes,  before  and  after  the  action  of  the  impulses,  we  have 

A  (o),  -  a)/)  =  X,  B  ((o2  -  W2r)  =  JT,  C  (a>3  -  a)/)  =  N,    (13) 


General  Expression  for  the  Vis  Viva  of  a  Body.       351 

where  A,  B,  C  are  the  principal  moments  of  inertia. 
Without  having  recourse  to  Art.  209,  we  may  deduce  equa- 
tions (13)  directly  from  (18),  Art.  204,  and  (20),  Art.  205, 
by  the  method  of  Ex.  2,  Art.  213. 

From  equations  (12)  and  (13)  it  appears  that  an  impulse 
whose  direction  passes  through  the  centre  of  inertia  of  a 
free  rigid  body  produces  a  motion  of  translation  only,  whereas 
an  impulse  not  passing  through  the  centre  of  inertia  pro- 
duces both  a  translation  and  a  rotation. 

266.  General  Expression  for  the  Vis  Viva  of  a 
Body. — As  the  motion  of  a  body  relative  to  one  of  its  points 
must  always  consist  of  a  rotation  round  some  axis  through 
the  point,  it  follows,  from  Art.  134,  that  if  a  body  be  free, 


where  %)l  is  the  mass  of  the  body  ;  Vth.e  velocity  of  its  centre 
of  inertia;  I  the  moment  of  inertia,  and  w  the  angular 
velocity,  round  the  instantaneous  axis  through  the  centre  of 
inertia. 

As  was  shown  in  Art.  263, 

I(o2  =  Aw2  +  Bu22  +  Cu2. 

Again,  if  a,  b,  c,  i,j\  k  be  the  moments  and  products  of 
inertia  for  the  centre  of  inertia,  round  three  rectangular  axes, 
which  are  parallel  to  fixed  directions  in  space,  and  wx,  o)y,  ws 
the  corresponding  angular  velocities  of  the  body, 

I(jj2  =  awx  +  bu)y2  +  cwz  -  2i(i)y(vz  -  %J(Dz(*>x  -  2kwx(t)y ; 

whence  we  have 

2mv2  =  WIV2  +  Am?  +  JW  +  CW  (14) 

-  %fl  V2  +  awx2  +  bwy2  +  cu)z  -  2hy  w~  -  2j(uz  wx  -  2kwxwy.  (15) 


352  Kinetics  of  a  Rigid  Body. 


Examples. 

1 .  A  free  body  is  set  in  motion  by  an  impulse.  If  the  initial  motion  be  a 
pure  rotation,  show  that  the  directions  of  the  impulse  and  of  the  instantaneous 
axis  of  rotation  are  principal  axes  of  a  section  of  the  momental  ellipsoid 
relative  to  the  centre  of  inertia. 

Since  the  initial  motion  is  a  pure  rotation,  the  initial  velocity  of  the  centre 
of  inertia  is  at  right  angles  to  the  direction  of  the  instantaneous  axis  of 
rotation.     The  above  statement  follows,  then,  from  Art.  264. 

2.  On  the  same  hypothesis  as  in  the  last  example,  show  that  the  instan- 
taneous axis  of  rotation  is  a  principal  axis  of  the  body,  at  the  point  in  which  it 
is  met  by  its  shortest  distance  from  the  line  of  direction  of  the  impulse  (see  Ex.  1, 
Art.  241). 

3.  If  different  impulses  applied  to  the  same  body  produce  velocities  of  ro- 
tation round  parallel  instantaneous  axes,  prove  that  in  general  these  axes  lie 
in  one  plane  containing  the  centre  of  inertia,  and  perpendicular  to  the  lines  of 
direction  of  the  impulses,  and  that  the  points  in  which  this  plane  meets  these 
lines  lie  on  a  straight  line. 

4.  If  in  the  preceding  example  the  instantaneous  axes  are  parallel  to  a 
principal  axis  through  the  centre  of  inertia,  prove  that  the  lines  of  direction  of 
the  impulses  lie  in  the  corresponding  principal  plane  at  the  centre  of  inertia. 

The  theory  of  the  centre  of  percussion,  given  in  Art.  235,  may  be  collected 
from  Examples  2,  3,  4. 

5.  A  body  is  moving  freely  :  under  what  circumstances  can  it  be  brought  to 
rest  by  an  impulse,  and  what  must  be  the  magnitude  and  position  of  the 
impulse  ? 

The  direction  of  the  impulse  must  be  opposite  to  that  of  the  velocity  of  the 
centre  of  inertia,  and  its  magnitude  must  be;  equal  to  the  momentum  of  trans- 
lation of  the  body.  Again,  the  moment  of  the  impulse  round  the  centre  of 
inertia  must  be  equal  and  opposite  to  the  couple  of  principal  moments.  Hence 
the  magnitude  and  position  of  the  impulse  are  determined,  and  the  motion  of 
the  body  must  be  such  that  the  momentum  axis  is  perpendicular  to  the  direction 
of  motion  of  the  centre  of  inertia. 

6.  A  free  body  is  set  in  motion  by  an  impulse  of  given  magnitude,  and  pass- 
ing through  a  given  point  P  of  the  body ;  find  the  directions  of  the  impulse  for 
which  the  initial  vis  viva  of  the  body  is  a  minimum,  and  for  which  it  is  a  maxi- 
mum. 

Since  the  impulse  is  given  so  is  the  velocity  V  of  the  centre  of  inertia ;  but 
the  total  vis  viva  1T=  m  V2  +  S,  where  S  is  the  vis  viva  of  the  motion  relative 
to  the  centre  of  inertia ;  hence  T  is  a  minimum  when  S  is  zero,  i.  e.  when  the 
direction  of  the  impulse  passes  through  the  centre  of  inertia. 

Again,  the  direction  of  the  impulse  for  which  S  is  a  maximum  is  found  as  in 
Ex.  9,  Art.  264,  and  when  S  is  a  maximum  so  likewise  is  T. 

7.  A  free  body  is  set  in  motion  by  an  impulse  whose  magnitude  and  perpen- 
dicular distance  from  the  centre  of  inertia  of  the  body  are  given  ;  find  the  direc- 
tion of  the  impulse  so  that  the  initial  vis  viva  of  the  body  may  be  a  maximum. 

Here  S  must  be  a  maximum,  and  therefore,  as  in  Ex.  1,  Art.  264,  the 
impulse  must  lie  in  a  plane  passing  through  the  centre  of  inertia  and  perpen- 
dicular to  the  axis  of  least  inertia  of  the  body. 


Equations  of  Motion  of  a  Body  having  a  Fixed  Point. '   353 


267.  Equations  of  Motion  of  a  Body  [having  a 
Fixed  Point. — In  the  case  of  continuous  forces,  if  Gx,  Gyy 
Gz  be  the  moments  of  the  applied  forces  round  the  space 
axes,  the  equations  of  motion  are  (25),  Art.  210, 


dRx  _         dm 

dt  -**-  It 


'-a.   ~-gz. 


'y> 


dt 


(16) 


We  may  substitute  for  Hx,  Hy.  and  Hz  in  these  equations 
their  values  given  by  (1),  or  by  (4).  If  we  make  the  former 
substitution  we  obtain 


dt 


\Ctu>x     ~  &*> 


■J«>* 


Gx~) 


COj 


Jt[~J<»X-   t<»y 


+  bu>y  —  i(ox)  =  Gv  y 
+  c<»z)  =  Gz) 


(17) 


In  the  case  of  homogeneous  spheres,  as  also  in  that  of  the 
initial  motion  of  a  body  starting  from  rest,  these  equations 
are  sometimes  useful ;  but  since  in  general  a,  k,  J,  &c.  vary 
with  the  time,  it  is  usually  necessary  to  reduce  equations  (17) 
to  a  more  manageable  form. 

If  we  substitute  for  Hx,  Hyi)  Hz  in  (16)  their  values  given 
by  (4)  and,  after  performing  the  differentiations,  suppose  the 
space  axes  to  coincide  with  the  instantaneous  positions  of  the 
principal  axes  of  the  body  at  0,  we  have  by  (6),  Art.  255,  and 
(9),  Art.  256,  remembering  that  in  this  case  a2,  aly  bly  £>,,  cl9 
c2,  are  each  zero,  and  that  ax  =  b%  ■  c3  =  1, 

dHx         dwx  n. 

dHy      ^dujz     in      A. 

-df=BTt-{0-A^^> 


dHz 

dt 


L~di 


{A.  —  B)  u)i  a>2 ; 


whence,  if  L,  M,  N  be  the  moments  of  the  applied  forces 

2  A 


354 


Kinetics  of  a  Rigid  Body. 


round  the  principal  axes  at  0,  the  equations  of  motion  be- 
come 

AC^-{B-C)w2w3  =  L 


dco2 
Blli 


(C-  A)wzMl  =  m   y 


C~-{A-B)u>lw2  =  N 
etc 


(18) 


Equations  (18)  are  due  to  Euler,  and  are  called  by  his 
name. 

268.  Equations  of  Motion  of  a  Free  Body. — If  we 

denote  the  mass  of  the  body  by  2D?,  we  have 

»?-«•  ^S=sF'  »£-**  (19) 


where  x,  y,  z  are  the  coordinates  of  the  centre  of  inertia  re- 
ferred to  any  three  rectangular  axes  fixed  in  space ;  and  "2X, 
S  F,  SZ  are  the  sums  of  the  components  of  the  applied  forces 
parallel  to  these  axes. 

Again,  since  the  motion  of  the  body  relative  to  its  centre 
of  inertia  is  the  same  as  if  that  point  were  fixed  in  space 
(Art.  209),  we  have 


ACl~  -  [B  -  C)  tt2wz  =  L 


dd)2 
~di 

d(i)2 
~di 


{C-A)wzwl  =M  y, 


-{A-B)muz  =  iV 


(20) 


where  o»i,  w2,  w3  are  the  angular  velocities  ;  A,  B,  C  the 
moments  of  inertia;  and  L,  3£,  iV  the  moments  of  the 
applied  forces  round  the  three  principal  axes  of  the  body 
at  the  centre  of  inertia. 


Examples.  355 

Instead  of  equations  (20)  we  may  use  (17),  the  axes  being 
parallels  through  the  centre  of  inertia  to  directions  fixed  in 
space. 

As  in  the  case  of  impulses,  equations  (20)  may  be  deduced 
directly  from  Art.  204  by  the  method  of  Ex.  2,  Art.  213. 

From  equations  (19)  and  (20),  it  appears  that  a  force 
whose  direction  passes  through  the  centre  of  inertia  of  a  free 
body  produces  a  motion  of  translation  only,  whereas  a  force 
not  passing  through  the  centre  of  inertia  produces  both  a 
translation  and  a  rotation. 

Examples. 

1.  A  body  is  given  a  rotation  round  a  principal  axis  through  its  centre  of 
inertia,  and  is  acted  on  by  a  couple  having  this  line  for  its  axis.  Show  that  the 
body  will  continue  to  revolve  round  the  axis  of  initial  rotation. 

2.  One  end  of  a  uniform  rod  rests  on  a  horizontal  plane  and  against  a  vertical 
wall ;  the  other  rests  against  a  parallel  vertical  wall.  All  the  surfaces  being 
smooth,  if  the  rod  slips'down,  determine  the  motion. 

Take  the  intersection  of  the  horizontal  and  vertical  planes  passing  through 
the  first  end  of  the  rod  for  axis  of  x,  and  a  vertical  plane,  at  right  angles  to  the 
-walls  and  passing  through  the  initial  position  of  the  centre  of  inertia  of  the  rod, 
for  the  plane  of  yz,  the  axis  of  z  being  vertical. 

Let  fi  be  the  angle  which  the  rod  at  any  time  makes  with  the  axis  of  y,  2a 
its  length,  2b  the  distance  between  the  walls,  x\,  yi,  z\ ;  x2,  y2,  z2  ;  and  x,  y,  z 
the  coordinates  of  the  two  extremities,  and  of  the  centre  of  inertia  of  the  rod. 
Then 

y\  =  0,     Z\  =  0,     y2  =  2b,     y  m  §  {yl  +  y2)  =  b  : 

also  y  =  a  cos  fi,  whence  cos  $  =  -  ;  thus,  as  fi  is  constant,  the  motion  of  the  rod 
relative  to  its  centre  of  inertia  is  a  rotation  round  the  axis  of  y,  whose  ampli- 
tude at  any  time  may  be  denoted  by  <p.     Again,  as  m  —  =  0,  and  as  the  initial 

value  of  —  is  zero,  it  is  zero  throughout  the  motion  ;  also,  since  y  is  constant, 

dy 

—  =  0  ;  whence  the  equation  of  vis  viva  is 

m (&<j>*+P)  =  2mg{z0-z). 

70      tf2  sin2j8      _  .    „ 

Now  k2  =  — ,     z  =  a  sin  j8  cos  cp, 

3 

whence,  as  the  initial  value  of  <p  is  zero,  we  obtain 

'6a 

(1  +  3  sin24>)  <p-  =  (1  -  cos  (f>). 

vV  -  ft5 
Also,  .T2  =  a  sin  £  sin  cp,  which  determines  the  position  of  the  upper  end  of  the 
rod  when  (p  is  known. 

2  A  2 


356  Kinetics  of  a  Rigid  Body. 

3.  A  heavy  body  is  supported  in  equilibrium  by  two  strings :  one  is  cut ;  find 
the  initial  tension  of  the  other. 

The  two  strings  and  the  centre  of  inertia  G  of  the  body  lie  at  first  in  the 
same  vertical  plane  ;  let  this  plane  be  that  of  yz,  the  axis  of  z  being  vertical,  and 
its  positive  direction  downwards,  and  let  the  origin  be  the  point  0  to  which  the 
uncut  string  is  attached.     (See  figure,  p.  293.) 

Let  I  be  the  length  of  the  string  OA,  and  h  the  distance  AG,  the  direction 
cosines  of  OA  being  a,  /3,  y,  those  of  AG,  A,  /j.,  v ;  then,  if  x,  y,  z  be  the  co- 
ordinates of  G,  we  have  y  =  10  +  hp,  z  =  ly  +  hv  ;  and,  if  T  be  the  tension  of 
the  string,  and  m  the  mass  of  the  body,  the  equations  of  motion  of  G  are 

mx  =  —  To,     my  =  —  Tfi,     m'z  =  my  -  Ty. 

Differentiating  the  expressions  for  y  and  z  twice,  substituting,  and  remembering 
that  the  initial  values  of  the  differential  coefficients,  with  respect  to  the  time,  of 
o,  &,  y,  A,  n,  v  are  each  zero,  we  get 

d2y       7  d2v  Ty 

Multiplying  the  first  of  these  equations  by  #,  the  second  by  y,  and  addiDg, 
we  have  initially 


I  m 


»{•(?)♦*  (3?) 

since  o2  4  &2  +  y2  =  1,  and  initially  o  =  0, 

(3MS0- 


and  therefore  j8 


Now  by  (6),  Art.  255,  since  A  is  zero  initially,  we  have 

d2/x  dwx      d2v         dwx 

dP=~V~dty     ~di2  =  ,M~d7' 


Hence  initially  h  ($v  —  y/i)  —  = yy. 

dt        in 


If  p  be  the  length  of  the  perpendicular  from  G  on  the  initial  position  of  the 
string,  this  equation  may  be  written 


T 

po>z  = yy- 

m 


Motion  of  a  Body  under  No  Forces. 


357 


Again,  if  a,  b,  c,  i,j,  k  be  the  moments  and  products  of  inertia  round  axes 
through  G  parallel  to  the  coordinate  axes,  we  have  initially 


(a<a»  -  kwy  -juz)  =  —  Tp, 
(—  kwx  +  bcay  —  ic>z)  =  0, 


j-  {-ji»x  -  iwy  +  CWz)  =  0. 

In  differentiating,  since  the  initial  values  of  o}x,  <t)y,  and  ws  are  each  zero,  a, 
&c,  may  he  treated  as  constants,  and  as  having  the  values  belonging  to  the 
initial  position  of  the  body.  We  have,  then,  for  the  initial  values  of  «,,  &c, 
and  T  the  equations, 

ad>z  —  kuy  —  jdz  =  —  Tp, 

—  kcbjc  +  bdly  —  i(iz  =  0, 

—  j(bx  —  ieoy  +  cci>s  =  0  ; 

whence,  if  A  be  the  determinant 

a    - k    -j     | 
k     b    -i 
j     -i      c 

we  obtain  Aux  =  -(be  -  i2)  Tp. 

Substituting  for  ux,  we  have  finally  for  To  the  initial  value  o    T, 
A70  ABCyo 


A  +  mp2,  (be  —  i2) 


mg- 


ABC  +  mp2  (be  -  v 


mg. 


269.  Motion  of  a  Body  round  a  Fixed  Point, 
under  the  Action  of  no  External  Force. — In  this 
case  equations  (18)  become 

in  which  we  shall  suppose  A  >  B  >  C. 

Multiply  the  first  by  o>i,  the  second  by  w2,  the  third  by  w8, 
add,  and  integrate,  and  we  have 


Cl»i 


+  Bus  +  CW  =  8. 


(21) 


358  Kinetics  of  a  Rigid  Body. 

Next  multiply  the  first  by  Aioi,  the  second  by  Boj2,  the 
third  by  Cwz,  add,  and  integrate,  and  we  have 

A2  wx2  +  B2  iv,2  +  C2  w32  =  H2.  (22) 

In  equations  (21)  and  (22)  S  and  H  are  constants. 
These  equations  could  have  been  obtained  directly  from  (3), 
Art.  261,  and  (7),  Art.  263,  by  articles  200  and  213. 

Again,  if  we  multiply  the  first  of  the  equations  obtained 

from  (18)  by  ^-,  the  second  by  -A  the  third  by  — s,  and  add, 
A  -t>  O 

we  get 

don        doj2        dm      (B-C      0  -  A      A  -  B\ 

»lW+t*i~df+"3W  =  \aT  +  ~b~  +  —^-)«^2«» 

doj        (A-B)(B-  C)(C-A) 
or  co-  = -^ «i<*<*. 

If  we  combine  the  two  equations  already  found  with  the 
equation 

wi2+  o>22  +  is)2  =  or, 

and  solve  for  cm2,  we  get 

1,      1,      1 
A,    B,     G 

A2    B2     C' 

'       ''.'-,             BO           (  ,    S(B+Q)-E>) 
whenoe       ""  =  (A-B)(A-C)  h BO 1  •  (23) 

Of  /  7?   i   /^  _   772 

If  we  denote   — ^~ by  Ai,  and  the  two  correspon- 

ding  quantities  by  A2  and  A3,  we  have 
ABO 


wi2=oj2BC{C-B)-S{C2-B2)  +  H2(C-B); 


My  lx>2  OJz 


(A-B){A-C)[B-C) 


v/{(^-A1)(A2-co2)(o>2-A3)); 


*  w =  v !  (Ai " w2)  (Az "  *2)  (As " "2) '  ■    (24) 


Motion  of  a  Body  under  No  Forces. 


359 


Again,  since 

AS  -H2  =  B{A-  B)u?  +  C{A-  C)u>z\ 

it  follows  that  AS  is  always  greater  than  IP ;  in  like  manner 
we  see  that  CS  is  less  than  H2.     Hence  we  see  at  once  that 
Ai,  A2,  and  A3  are  each  positive  quantities. 
Also,  we  have 


A3  —  Ai 


A-B 


[H2-CS)>    Aa-As  = 


B-C 


(AS-E2); 


ABCX~      ~~"     "'     "°     ABC 

therefore  A2  is  the  greatest  of  the  three  ;  also  Ai  -  A3  has  the 
same  sign  as  BS  -  H2,  and  this  depends  on  the  initial  con- 
ditions. 

Again,  since  on2,  o>22,  <d32  are  each  positive, 

to2  >  Ai,   u>2  <  A2,   o>2  >  A3. 

Hence  we  may  assume  either — 

(1),  w3  =  A1sin2<£  +  A2cos2<£,  or  (2)  a>2  =  A3sin2^  + A2cos2^. 

In  the  former  case,  if  we  select  the  negative  sign  of  the 

square  root  in  (24),  that  equation  gives 

d<j> 


dt 
where 


=  \/A2-A3-  (A3-  Ai)  sin20  =  ^/A2-A3  v^l-^sin2^,     (25) 


72     A2-At 

to     =  r- 


Hence,  by  (23),  we  get 
BO 


(A-B){A-C) 
AC 


A2 


-  (A2- Ai)cos20 


(A-B)(B-0) 

AB 
(A-0)(B-0) 

In  the  second  case,  we  get 


V-(A2-Ai)sins^ 


«3 


(X„-Xs)(l-A8sin» 


f  =  yx2  -  X,  Jl  -  J  sin'*, 


>•    (26) 

I 

I 

J 


(27) 


360 


and 


Kinetics  of  a  Rigid  Body. 


(X)V 


tjj%  = 


(-1 


£p^o)(Aj-Xl)(1_  Jsin^] 


AG 


0)j 


(A-B)(B-C) 

AB 
{A-C)[B-C) 


(X2  -  A3)  sin2^ 
(A2- A3)  cos2^ 


(28) 


It  is  obvious  that  <f>  and  if/  are  connected  by  the  equation 
sin  \\f  =  k  sin  (p. 

We  thus  see  that  when  either  $  or  if/  is  known  the 
values  of  wi,  o>2,  a>3  can  be  determined.  Also,  from  (25)  and 
(27),  we  see  that  <p  and  iff  are  at  once  expressible  in  terms 
of  t  as  elliptic  functions  of  the  first  kind. 

We  now  proceed  to  give  a  geometrical  representation  of 
the  angles  <£  and  iff. 

Let  x,  y,  z  be  the  coordinates  of  the  point  P  in  which  the 
momentum  axis,  at  any  instant,  intersects  the  surface  of  the 
ellipsoid  of  gyration, 


then,  if  iiJ  =  OP,  we  have 

Hence         ^J-^*-**-?-  /SSS;  (Ex.  8,  Art.  264) 
x  y  z        R 


B 


</ms 


>d     2/ 


v/9J?# 


W2,        S 


yms 


U>3« 


(29) 


Hence,  in  terms  of  0,  we  have 


JUB0 


(.1 -if)  (-4-0) 


<v/A2-  AiCOS0 


(30) 


I  B       I         ^JSO  /c — r-  .     . 


Motion  of  a  Body  under  No  Forces.  361 

And,  in  terms  of  ifr, 

»-Hal<4-w-Q)'*r:*da*\    (81) 

ABC 


[A-Oifi-o^-^™*  J 


These  equations  show  that  the  position  of  the  momentum 
axis  in  the  body  is  determined  when  either  0  or  ^  is  known. 
If  we  write  equations  (30)  in  the  form 

x  =  a  y/\2  -  Ai  cos  (p,  y  =  j3  <s/\%  -  Ai  sin  0, 
we  get 

*  +  t  =  x2  -  A,  32) 

a2        £2 

This  is  the  equation  to  the  projection  on  the  plane  of  xy  of 
the  curve  described  by  P  on  the  surface  of  the  ellipsoid  of 
gyration.  Next,  if  o-  be  the  angle  which  a  cyclic  plane  of 
the  ellipsoid  of  gyration  makes  with  the  plane  of  xy,  we 
easily  see  that 


a    fg  -  c2        U 
byjcf-c?  ~  \B 


{B  -  C)       a 


From  this  it  follows  that  lines  parallel  to  the  axis  of  z  will 
project  the  ellipse  (32)  into  a  circle  on  the  cyclic  plane.  This, 
in  fact,  is  a  well-known  theorem  in  surfaces  of  the  second 
degree,  since  the  locus  of  P  is  a  sphero-conic.  (Ex.  8, 
Art.  264.) 

In  like  manner  it  follows  from  (31)  that  the  projection  of 
this  sphero-conic  on  the  cyclic  plane  by  lines  parallel  to  the 
axis  of  x  is  another  circle. 

These  circles  in  the  cyclic  plane  are  exhibited  in  the 
accompanying  figure,  in  which  OF  is  the  mean  axis  of  the 
ellipsoid.  If  we  suppose  \,  >  A3,  i.e.  BS  >  H\  then  it  is  easily 
seen  that  the  inner  circle  is  the  projection  by  lines  parallel 
to  the  axis  of  z.  Hence,  if  M  and  M'  be  the  positions  of  the 
projections  of  P  at  any  instant,  we  shall  have  MOQ  =  <p, 
and  M'OQ'  =  \p.  This  construction  for  the  position  of  the 
momentum  axis  in  the  body  is  due  to  Mac  Oullagh. 


362 


Kinetics  of  a  Rigid  Body. 


From  Article  213  it  appears  that  the  direction  of  the 
momentum  axis  in  space  is  invariable. 

It  should  be  observed  that  these  results  also  hold  good 


for  the  motion  of  a  free  body  under  the  action  of  no  forces, 
provided  its  centre  of  inertia  be  taken  as  the  origin. 


Examples. 

1.  If  -ZT2  =  BS,  find  the  position  of  the  momentum  axis  at  any  instant. 
It  is  immediately  seen  that  in  this  case  the  momentum  axis  always  lies  in  a 
cyclic  plane  of  the  ellipsoid  of  gyration.     Also,  since 

Ai  =  A2,     we  have  Je  =  1. 

Hence  equation  (25)  becomes 

\{A-B){B-C)S 


— i—  =  a/a2  -  Ai  dt  =  hdt,       where  h 

COS(p 


ABC 


Hence 


and  we  get 


log  tan  f  -  +  I)  =  ht  +  constant, 


where  #0  is  the  initial  value  of  <p.  Hence  the  momentum  axis,  and  therefore  also 
the  instantaneous  axis  of  rotation,  tends  to  approach  without  limit  to  the  mean 
radius  of  gyration. 


Conjugate  Ellipsoid  and  Conjugate  Line.  363 

2.  Investigate  the  motion  if  the  initial  axis  of  rotation  he  very  close  to  the- 
least  axis  of  inertia. 

In  this  case  It  is  nearly  equal  to  c,  and  .-.  S2  -  CS  is  very  small :_  accord- 
ingly k  is  a  very  small  quantity,  and  we  get  approximately,  from  equation  (25), 


<f>  =  y\2-\3t+<l>o  =  ^^- ^ +  <P°' 

where  <£o  is  the  initial  value  of  (p. 

If  t\  denote  the  time  which  the  body  takes  to  revolve  round  its  axis  of  rota- 
tion (which  is  nearly  coincident  with  the  least  axis  of  the  ellipsoid  of  gyration), 
and  to  the  time  of  a  complete  revolution  or  oscillation  of  the  momentum  axis  round 
the  axis  of  z  ;  then 


ffc-Sr,     ?  J{B~Gl{»~G)h  =  2*,     R  =  ct     approximately. 
(j  6   A!  AH 

Hence,  approximately, 


vu- 


AB 

h  =  h 


[A  -C)(B-  C) 

If  B  —  Che  very  small,  h  will  he  very  large  in  comparison  with  t\. 

A  corresponding  result  may  be  obtained  when  R  nearly  =  a. 

This  investigation  would  be  applicable  to  the  Earth  if  its  axis  of  rotation 
were  nearly  but  not  exactly  a  principal  axis.  In  this  case  t\  would  be  the  length 
of  the  day.  The  total  attractions  of  other  bodies  are  supposed  to  pass  through 
the  centre  of  inertia  of  the  Earth. 

3.  If  two  of  the  principal  moments  of  inertia  of  the  body  be  equal,  prove  that 
— (1)  the  simultaneous  positions  of  the  momentum  axis  and  the  instantaneous 
axis  of  rotation  lie  in  a  plane  containing  the  axis  of  unequal  moment  of  inertia  ; 
(2)  the  instantaneous  axis  and  the  momentum  axis  describe  in  the  body  right 
circular  cones  whose  semi-  angles  are  i  and  7,  where 

C    H*-CS  ,    4     ,         A   H*-CS 

the  axis  of  unequal  moment  of  inertia  being  the  axis  of  z  ;   (3)  the  values  of 
«,  wi,  o>2,  W3  at  any  time  are  given  by  the  equations 


u\  =  u  sin  1  cos 


(A-  C  \  .    .   .    (A-G      _     \ 

f —  a>3 1  +  x  J 1    «2  =  -  »  sm  t  sin  I  — —  mi  +  XJ 


.  (A+  C)  S-  3°- 

o>3  =  o)  cos  »,  a»2  = - ,     where  x  is  an  arbitrary  constant. 

AG 


270.  Conjugate  Ellipsoid  and  Conjugate  line- 
When  a  body  on  which  no  external  force  is  acting  is  in 
motion  round  a  fixed   point,   the    squares   of   the   angular 


364  Kinetics  of  a  Rigid  Body. 

velocities  of  the  body  round  its  principal  axes  at  the  point 
must  fulfil  the  two  independent  linear  equations 

An?  +  B(o22  +  CV  -  S  =  0  =  0  | 

Any  other  linear  equation,  0'  =  0,  between  these  variables 
must  be  of  the  form  aQ  +  (3<P  =  0,  where  a  and  /3  are  con- 
stants, since  otherwise  each  angular  velocity  would  be  com- 
pletely determined.  Hence,  if  we  suppose  that  wi,  w2,  wj 
satisfy  also  the  two  equations 


(34) 


we  must  have  0'  =  i  (X0  -  <£),  $>'  =  y  (^0  -  <P),  where  i,  X, 
;,  jn  are  constants. 
Hence  we  get 

A'  =  iA  (X  -  A),  A'2  =jA  (p-A)\ 

Bf  =  iB  (X  -  £),  B'2  =  jB  [fi  -  S)  |  •  (35) 

a  =iC(\  -o),  a2  =jc(n-  c)) 

From  (35)  we  obtain 

AA^-A)2     B(X-B)2      C{\-C)2     j 


A'wS  +B'w22  +  CW 

-  S'  =  e'  =  0 

A'2*?  +  #W  +  C'W 

-H,2=&  =  0 

B  u-0 


,2  ' 


(36) 


whence,  by  a  well-known  property  of  equal  fractions,  we 
have 

A(\-A)2-B(\-B)2  _B(\-B)2-C(\-C)2       j 

A-B  B-C  "    y*v*n 

from  these,  by  performing  the  divisions,  we  get 

X  =  \{A  +  B  +  C).  (38) 


Conjugate  Ellipsoid  and  Conjugate  Line.  365 

Consequently,  from  (37)  we  obtain 

2(AB  +  BC  +  CA)  -  A2  -B2-  C2  =  i;,  (39) 

and  from  (36)  we  get 

[2(AB  +  BC+CA)-A2-B2-C2}fi  =  iABC.       (40) 
Finally,  we  have 

d'=±A(B+  G-A),  BrJB{C+A-B),(r=±C(,A+B-C)\ 

A  TIC1  \ 

S'=  i(XS  -  E%        B'2  =  i2  ^p  {fiS  -  H2)  ) 

where  A  and  /u  have  the  values  given  by  (38)  and  (40). 

It  appears  from  what  has  been  said  that  any  three  con- 
stants a,  b,  c  satisfying  two  equations  of  the  form 

aw2  +  b(v22  +  CM2  =  constant,  a2w2  +  62<u22  +  c2w2  =  constant, 

must  be  proportional  either  to  A,  B,  C,  or  to  A',  B*,  C 
Hence,  if  we  apply  to  A',  B\  C  a  transformation  similar 
to  that  which  has  been  applied  to  A,  B,  C  we  must  obtain 
quantities  proportional  to  A,  B,  C.  From  this  it  follows 
that  the  two  quadrics  E  and  E'  given  by  the  equations 

Ax2  +  By2  +  Cz2  =  K,  Ax2  +  B'if  +  C'z2  =  K\ 

are  each  derived  from  the  other  by  a  similar  process,  and  may 
therefore  be  called  conjugate.  Since  A,  B,  C  are  each  positive, 
and  such  that  the  sum  of  any  two  is  greater  than  the  third, 
it  follows  from  (38),  (40),  and  (41),  that  A\  B\  C,  A,  p,  8\ 
and  EL'2  are  each  positive.  Hence  we  infer  that  the  quadric 
E'  is  an  ellipsoid. 

If  /  be  the  intercept  made  by  the  conjugate  ellipsoid  E' 
on  the  instantaneous  axis  of  rotation  of  the  body,  p'  the 
perpendicular  from  the  fixed  point  on  the  tangent  plane  to 
Er  at  the  extremity  of  r',  and  §'  the  angle  between  /  and  p\ 
it  can  be  proved  in  the  same  manner  as  in  Art.  264,  that 


,      S'      ,      \K'         ,   VK'S' 
w  cos  cj>  =  -g,,  r  =   /-£,-  w,  p  =—-g, — 

The  perpendicular  to  the  tangent  plane  to  i?'at  the  extremity 


366  Kinetics  of  a  Rigid  Body. 

of  r   corresponds  to  the  momentum  axis  in  the  momental 
ellipsoid,  and  is  called  the  conjugate  line. 

This  Article  and  the  following  Examples  are  taken  from 
a  Paper  by  Dr.  Eouth  in  the  Quarterly  Journal  of  Pure  and 
Applied  Mathematics  for  1888. 

Examples. 

1.  If  a  body  on  which  no  external  force  is  acting  be  moving  round  a  fixed 
point  0,  and  a  quadric,  having  as  axes  the  principal  axes  of  the  body  at  0,  be 
such  that  the  intercept  which  it  makes  on  the  instantaneous  axis  of  rotation  at 
any  time  is  proportional  to  the  angular  velocity,  and  that  the  perpendicular 
from  0  on  the  tangent  plane  at  the  extremity  of  this  intercept  is  constant,  the 
quadric  must  be  either  the  momental  or  the  conjugate  ellipsoid. 

2.  If  P  be  ?  point  on  the  conjugate  line  at  a  constant  distance  R  from  the 
fixed  point  0,  and  Q  the  point  of  the  body  which  coincides  at  the  instant  with 
P,  prove  that  the  velocity  of  P  is  double  that  of  Q,  and  that  the  directions  of 
these  two  velocities  coincide. 

Let  x,  y,  z  be  the  coordinates  of  Preferred  to  the  principal  axes  at  0;  u,  v,  to 
its  space' velocities  parallel  to  these  axes ;  and  u',  v  ,  w'  those  of  Q  ;  then 

ll'  =  (02 Z   -  0)3]/,       v' —  <*z%  —  WlZ,       W' =  (till/  —  (02%, 

a  =  x+  ?/,  v  =  y  +  v',  tv  =  z  +  to'. 

Now, 

H'x  =  RA'wi,     R'y  =  P£'w2,     H'z  =  PC'w2 ; 

hence,  by  (41),  we  have 

x  =  ^r,A(B  +  C-A)d,u 

iR 
and  W  =  —  (B  -  C)  (B  +  C-  A)  co2  o>3 ; 

whence,  by  Euler's  equations,  Art.  267,  we  obtain  x  =  u',  and  therefore  u  =  2u  ; 
and  in  like  manner  v  =  2v'}   w  =  2iv'. 

3.  Determine  the  motion  of  the  conjugate  line  in  space. 

Let  9  be  the  angle  between  the  conjugate  line  OP  and  the  invariable  line  or 
momentum  axis  OZ,  \|/  the  angle  which  the  plane  ZOP  makes  with  a  fixed 
plane  passing  through  OZ,  (p  and  <p'  the  angles  made  with  OZ  and  OP  by  the  in- 
stantaneous axis  of  rotation  of  the  body  ;  also  let  n  be  the  component  round  OZ 
of  the  angular  velocity  of  the  body,  and  n'  its  component  round  OP ;  then 

S    ■  ,  ,      8' 

n  =  ca  COS  <p  =  — ,   ft  =  <a  COS  (p   =  —. 
si  Ji 

By  considering  the  motion  of  a  point  of  the  body  situated  at  the  instant  on 
OP,  it  is  plain  from  Ex.  2,  that  the  angular,  velocity  of  the  body  round  an  axis 
perpendicular  to  OP  in  the  plane  ZOP  is  \  sin  dty.     Hence 

ft  =  \  sin2  dip  +  ft'  cos  6, 


Stress  Exerted  by  a  Body  on  a  Fixed  Point  367 

(S      S'  \ 

--— ^osej.  («) 

Again,  the  whole  velocity  of  a  point  of  the  body  at  the  unit  distance  from  0  on. 
the  line  OP  being  a>  sin  «£',  we  have,  by  Ex.  2, 

\  (sin2  6ip  +  6°-)  =  co3  sin2  <p', 
that  is 

sin*  0^+#  =  4  ("*-—  )•  (*) 

"We  have  now  to  express  w2  in  terms  of  0,  which,  can  be  done  as  follows  : — 
Expressing  cos  9  in  terms  of  the  direction  cosines  of  OP  and  OZ,  we  have 

EE'  cos  0  =  AA'  an2  +  BB'wi-  +  C'C"w32 

=  tA  (^2 an2  +  -S22  co22  +  C-  m-)  -  i  {A*  an2  +  B*  w22  +  C3  a>32). 

Hence  we  get 

H  H ' 
A*  «i2  +  2?3a>22  +  C3  m2  =\E —  cos  9. 

If  we  combine  this  equation  with  (33),  and  solve  for  <av2,  we  obtain 

p ri  1  TTTT' 

{A  -  B){B-  C){0-  A)wv  = —  \SBC  +  E*(A-K)--^-  cosi 

From  this  and  the  similar  expressions  for  a>22  and  a>32  we  get 

TTTT' 

ABCar  =  S{AB  +  BC+  CA)  -±EZ(A  +  B+  C) —  cos  6. 

Substituting  the  value  for  or  given  by  this  equation  in  (b),  we  obtain 
sin20i^+  6*=-^--{S(AB  +  BC+CA)-±E2(A  +  B+C)} 

From  (a)  and  this  equation  0  and  \p  can  be  obtained  by  quadratures. 

271.  Stress  Exerted  by  a  Body  on  a  Fixed  Point. 

— In  order  to  determine  the  force  exerted  by  a  fixed  point  on 
a  body  we  have  only  to  consider  the  point  as  replaced  by 
a  force,  whose  components  are  X0,  T0,  Z0,  passing  through  it. 
We  may  then  consider  the  body  as  free,  and  we  have,  by 
Article  268, 

dr 

with  two  similar  equations. 

But  as  the  body  is  rotating  round  the  origin,  if  we  sup- 
pose the  axes  fixed  in  space  to  coincide  at  the  instant  under 


368 


Kinetics  of  a  Rigid  Body. 


consideration  with  the  principal  axes  through  the  origin,  we 
have 

d2x        _  dioz      _  d(jjz  ._  k      ,    2        2v  . 

=  -  y  —  +  z  -jj-  +  wi  (yah  +  S(u3)  -  (wz  +  ivs )  x. 


df 


dt 


dt 


Substituting  for  ~  and  -£r  from  Euler's  Equations,  we 


dt 


dt 


n   _M 


+  Wl(5  +  C-A)  U~  +  S  ^)"  K  +  "a2)  x 


get, 

<#2  = 

Now,  let  ft,  ft,  ft  be  the  components  of  the  stress  on  the 
fixed  point  at  any  time,  in  the  directions  occupied  at  the 
instant  by  the  principal  axes  of  the  body,  then  ft  =  -  X0, 
and  therefore 

^  =  2r-3Ti[-?j+i^+co2(C+^-JB)^  +  l9-)-(c32+^)^  >,(42> 
^  =  2^-^[-|f+^+W3^  +  ^-^)(^+^2)"(&,l2  +  &,22)^l 

where  £,  »?,  X  are  the  coordinates  of  the  centre  of  inertia 
referred  to  the  principal  axes  through  the  fixed  point,  and 
are  absolute  constants :  2X  is  the  sum  of  the  components 
of  the  applied  forces  parallel  to  one  of  these  axes,  and  L  the 
moment  round  it  of  the  same  forces.  2X,  S  F,  SZ,  L,  M,  N 
are  in  general  variable  with  the  time. 

In  like  manner  if  ft,  ft,  ft  be  the  impulses  arising  from 
the  instantaneous  stresses  "exerted  by  a  body  on  a  fixed  point, 
in  consequence  of  the  action  on  the  body  of  any  system  of 
impulses,  we  obtain,  by  Arts.  255  and  265, 


Centrifugal  Couple,  369 

272.  Centrifugal  Couple. — If  a  body  have  a  fixed 
point  0,  the  change  produced  in  its  angular  velocity  round 
one  of  its  principal  axes  at  0  in  the  element  of  time  dt  is 
given,  (18),  Art.  267,  by  the  equation 

Adu>i  =  (B  -  C)  (Diaz dt  +  Ldt. 

The  first  term  on  the  right-hand  side  of  this  equation 
results  from  the  angular  velocities  already  existing  round  the 
other  two  axes.  In  consequence  of  these  velocities  each  point 
of  the  body,  in  virtue  of  its  connexions  with  the  other  points, 
exerts  a  force  on  the  entire  body.  These  forces  are  in  fact 
the  centrifugal  forces  resulting  from  the  motion  of  the  body, 
and  their  moments  L\  M\  N'  round  axes  fixed  in  space  may 
be  determined  directly  as  follows  : — 

Let  a,  j3,  y  be  the  angles  which  the  instantaneous  axis  of 
rotation  makes  with  the  axes  of  coordinates ;  p  the  perpen- 
dicular distance  from  this  axis  to  any  point  xy%  of  the  body ; 
q  the  intercept  between  the  origin  and  the  foot  of  p  ;  r  the 
radius  vector  to  the  point  xyz  ;  and  w  the  angular  velocity  of 
the  body  round  the  instantaneous  axis.  The  centrifugal 
force  at  the  point  xyz  is  mpu>2  acting  along  p  ;  and  the 
component  of  this  force  along  the  axis  of  x  is  mco2  multiplied 
by  the  projection  of  p. 

If  we  project  the  triangle  formed  by  rpq  on  the  axis  of 
xf  we  have 

projection  of  p  =  projection  of  r  -  projection  of  q  =  x  -  q  cos  a, 

and  q  =  x  cos  a  +  y  cos  f5  +  z  cos  y ; 

hence  the  centrifugal  force  along  axis  of  x 

=  mco2  [x  -  (x  cos  a  +  y  cos  /3  +  z  cos  y)  cos  a) 

=  mia*  {x  (cos2 3  +  cos2y)  -  y  cos  a  cos/3  -z  cos  a  cos  y J 

=   m{X  ((Dy*  +    (DZ2)  -   yWX  il)y  -  Z(DX  (1)Z), 

remembering  that 

wx  =  <»>  COS  a,      (i)y  =  (i)  COS  ]3,      wz  =  u)  COS  y. 
2  B 


370  Kinetics  of  a  Rigid  Body. 

In  like  manner  for  the  force  along  the  axis  of  y,  we  have 
m  \y  (u)z2  +  tax)  ~  z<*>y  wz  —  xiiiytox}  j 
and  for  that  along  the  axis  of  s, 

in  \z  wx2  +  wy~)  -  xwz ivx  -  yiDZ  (jjy) ; 

whence,  taking  moments  round  the  axis  of  x,  and  integrating 
through  the  entire  body,  we  obtain 

L'  =  [wy  -  idz2)  jyzdm  +  mv  wz  J  (z*  -  y1)  dm  -wzwzj  xydm 

+  d)xwyjxzdm.        (44) 

If  we  now  suppose  the  axes  to  coincide  with  the  instan- 
taneous positions  of  the  principal  axes  of  the  body,  every 
term  in  11  vanishes  except  wywz  j  (z2  -  y%)  dm,  and  we  get 

Lf  =  {B-C)<v2t»3.  (45) 

Accordingly,  the  couple  whose  components  round  the 
three  axes  are  [B  -  C)  w2  (i>3>  &c.,  is  called  the  centrifugal 
couple. 

The  axis  of  the  centrifugal  couple  is  at  right  angles  to  the 
axis  of  principal  moments,  and  to  the  axis  of  rotation. 

For  the  direction  cosines  of  the  axis  of  the  centrifugal 
couple  are  proportional  to 

(B  -  C)  Mi  w3>    [C  -  A)  wo  w„     [A  -  B)  h)i  wo ; 

whence  it  is  seen  at  once  that  the  conditions  for  its  being 
perpendicular  to  the  two  other  lines  are  fulfilled. 

If  a  central  section  of  the  momenta!  ellipsoid  be  taken 
passing  through  the  instantaneous  axis  of  rotation  and  the  axis  of 
the  centrifugal  couple^  these  two  lines  coincide  with  the  principal 
axes  of  this  section. 

The  lines  in  question  are  at  right  angles,  and  one  is 
parallel  to  the  tangent  plane  through  the  point  where  the 
other  intersects  the  ellipsoid. 


Examples.  371 

273.  Hotion  of  a  Free  Body  relative  to  its  Centre 
of  Inertia. — As  the  equations  for  determining  the  motion 
of  a  body  relative  to  its  centre  of  inertia  are  the  same  as 
if  the  centre  of  inertia  were  a  fixed  point,  the  theorems 
of  Arts.  264  and  272,  in  reference  to  the  instantaneous  axis 
of  rotation,  the  axis  of  the  centrifugal  couple,  and  the  axis 
of  principal  moments,  hold  good. 

Examples. 
Motion  of  a  Body  unacted  on  by  Force. 

1.  The  angular  velocity  at  any  instant  is  proportional  to  the  intercept  on  the 
instantaneous  axis  of  rotation  through  the  centre  of  inertia  cut  off  by  the 
momental  ellipsoid. 

The  velocity  of  the  centre  of  inertia  is  constant  as  well  as  the  whole  vis  viva. 
Hence  the  vis  viva  of  the  motion  relative  to  the  centre  of  inertia  is  constant,  and 
therefore,  (10),  Art.  264,  w  is  proportional  to  r. 

2.  The  component  of  the  angular  velocity  round  the  momentum  axis  through 
the  centre  of  inertia  is  constant.     See  (9),  Art.  264. 

3.  If  a  tangent  plane  be  drawn  to  the  momental  ellipsoid  at  its  point  of 
intersection  with  the  instantaneous  axis  of  rotation  through  the  centre  of  inertia, 
the  distance  of  this  plane  from  the  centre  is  constant. 

This  follows  from  (LI),  Art.  264. 

If  a  body  have  a  fixed  point,  the  results  of  the  preceding  examples  hold  good, 
the  fixed  point  being  substituted  for  the  centre  of  inertia. 

4.  A  body  moves  round  a  fixed  point :  give  a  geometrical  representation  of 
the  motion. 

The  momental  ellipsoid  relative  to  the  point  rolls  on  a  plane  fixed  in  space,  so 
that  the  line  joining  the  centre  to  the  point  of  contact  is  always  the  instan- 
taneous axis  of  rotation. 

5.  A  body  is  moving  round  a  fixed  point ;  find  the  locus  of  the  instantaneous 
axis  of  rotation  in  the  body. 

Since  — ,  —  ,  —  are  its  direction  cosines  referred  to  the  principal  axes  through 

CO         CO         CO 

the  point,  its  locus  is  the  cone 

A{W  -  AS)  x*  +  B  (W  -  BS)  y*  +  C(H*  -  CS)  &  =  0. 

6.  Find  the  locus  of  the  momentum  axis  in  the  body. 
Its  locus  is  the  cone 

r2        v2        z2  g 

A  B      J  0 

Hence  the  curve  traced  out  by  this  line  on  the  ellipsoid  of  gyration  is  a 
sphero-conic,  as  already  stated  in  Art.  269. 

2B  2 


372  Kinetics  of  a  Rigid  Body. 

7.  Determine  the  curve  traced  out  on  the  momental  ellipsoid  by  the  instan- 
taneous axis. 

The  equations  of  the  curve  are  got  by  combining  the  equations  ot  tne  ellip- 
soid with  that  of  the  cone  given  in  Ex.  5  ;  they  are,  therefore, 

Ax>  +  By2  +  Cz2  =  K,    AW  +  Bhf  +  C  V  =  — . 

This  curve  is  called  the  polhode. 

The  curve  traced  out  on  the  fixed  plane,  by  the  point  of  contact,  is  called 
the  herpolhode. 

8.  The  projections  of  the  polhode  on  the  planes  perpendicular  to  the  axes  of 
greatest  and  least  moment  of  inertia  are  ellipses.  Its  projection  on  the  plane 
perpendicular  to  the  remaining  principal  axis  is  a  hyperbola. 

This  appears  at  once  by  eliminating  x,  y,  z  successively  from  the  two  equa- 
tions of  Ex.  7,  remembering  that  A>B  >C. 

9.  In  what  case  does  the  hyperbola  become  a  pair  of  straight  lines  ? 
HH~=BS.     (See  Ex.  1,  Art.  269.) 

10.  If  the  body  be  free,  give  a  geometrical  representation  of  the  motion.    (See 

The  momental  ellipsoid  relative  to  the  centre  of  inertia  rolls  on  a  plane  at  a 
constant  distance  from  the  centre  of  inertia  and  parallel  to  a  plane  fixed  in  space, 
the  instantaneous  axis  of  rotation  being  the  line  joining  the  centre  of  inertiato 
the  point  of  contact,  whilst  the  whole  system  moves  with  uniform  velocity 
parallel  to  a  fixed  direction. 

11.  Show  that  the  herpolhode  lies  between  two  circles  the  squares  of  whose 
radii  are 

Kf         S*\  3      K(  S2\  Kl  S-\ 

-^(A2--),     and      _^Ai--j,or    j^-^J 

according  as  (see  Art.  269)  Xi  is  greater  or  less  than  A3- 

If  p  be  the  distance  from  the  point  of  contact  to  the  foot  of  the  perpendicular 
on  the  fixed  plane,  we  have 

7T9  TT  K I  S2  \ 

?  =  r^p>\  but^  =  ^,andr2  =  -a>MArt.264);   /.^-  -^  a>*  -  -,). 

But,  Art.  269,  co2  >M,     »2<A2,     &>2>A3.  . 

Hence  the  greatest  and  least  values  of  p2  are  comprised  between  the  limits- 
stated  above. 

12.  If  a  body  be  rotating  round  a  fixed  point,  or  a  free  body  round  its  centre  of 
inertia,  the  couple  resulting  from  centrifugal  forces  lies  in  the  plane  containing 
the  momentum  axis  and  the  instantaneous  axis  of  rotation,  and  its  magnitude 
is  Eu>  sin  cp,  or  S  tan  <p,  where  <p  is  the  angle  between  the  instantaneous  and  the 
momentum  axes. 


Examples.  373 

The  components  of  the  couple  resulting  from  centrifugal  forces  are  (Art. 
272) 

(B  -  C)  Ci>2  &>3,        (0—  A)  0>3  COl,        (j4  —  B)  £01  C02, 

or  w«2  (b2  —  c-)  cos  /3  cos  7,  wco2  (c2  —  a2)  cos  7  cos  a,  mca2  (a2  —  b2)  cos  a  cos  /3  ; 

where  a,  )8,  7  are  the  angles  made  by  the  instantaneous  axis  of  rotation  with  the 
principal  axes  of  the  body,  and  a,  b,  e  are  the  semi- axes  of  the  ellipsoid  of 
gyration.  Jf  ;;  be  the  perpendicular  from  the  origin  on  the  tangent  plane  to 
the  ellipsoid  of  gyration  at  the  point  x'y'z'  where  it  is  met  by  the  momentum 
axis  B,  double  the  projection  of  the  triangle  formed  by  the  origin,  x'y'z,  and 
the  foot  of  jo,  is 

p  (x'  cos  £  -  y'  cos  a) ,     or     (a2  —  b2)  cos  a  cos  (3, 

and  double  the  area  of  the  same  triangle  is  Bp  sin  c/> ;  therefore  by  Ex.  5,  7,  8, 
Art.  264,  we  have  the  required  result. 

13.  If  a  tangent  plane  be  drawn  to  the  ellipsoid  of  gyration  at  the  point 
where  it  is  met  by  the  axis  of  the  centrifugal  couple,  the  perpendicular  on  this 
tangent  plane  is  the  axis  of  the  rotation  produced  by  the  centrifugal  couple. 

L',  M'y  N'  being  the  components  of  the  centrifugal  couple,  and  5coi,  5«2, 
S0.3,  the  rotations  produced  by  it  considered  alone,  we  have,  from  Euler's  equa- 
tions, 

A5m  =  L'dt,     B8w2  =  M'dt,     C5«3  =  K'dt ; 

but  these  equations  are  of  the  same  form  as  those  connecting  the  instantaneous 
axis  with  the  components  of  the  couple  of  principal  moments  ;  therefore,  &c. 

It  follows  from  this,  that  the  axis  of  rotation  produced  by  the  centrifugal 
couple  is  at  right  angles  to  the  momentum  axis  ;  for  {see  Fig.,  Ex.  16)  if  OB  be 
the  momentum  axis ;  OP  the  instantaneous  axis  of  rotation ;  OB'  the  axis  of 
the  centrifugal  couple,  and  OB'  the  axis  of  the  centrifugal  couple  rotation  ;  OR' 
being  at  right  angles  to  OB  (Ex.  12),  is  conjugate  to  OB  :  hence  OB  is  parallel 
to  the  tangent  plane  through  B' ,  and  therefore  at  right  angles  to  OB' .  Also,  OB 
and  OBJ  are  the  principal  axes  of  the  section  of  the  ellipsoid  made  by  their 
plane. 

14.  The  intercept  on  the  momentum  axis  cut  off  by  the  ellipsoid  of  gyration 
is  of  constant  length  (Ex.  8,  Art.  264). 

15.  The  motion  of  the  momentum  axis  in  the  body  consists  of  a  series  of 
rotations,  the  axis  of  each  rotation  being  at  right  angles  both  to  the  momentum 
axis  and  the  centrifugal  couple  axis,  and  the  magnitude  of  the  rotation  being 
equal  and  opposite  to  the  rotation  of  the  body  round  the  same  axis. 

The  centrifugal  couple  tends  at  each  instant  to  alter  the  position  of  the 
momentum  axis,  since  the  new  moment  of  momentum  is  the  resultant  of  the 
principal  couple  at  the  beginning  of  the  instant  and  the  momentum  produced  by 
the  centrifugal  couple  during  the  instant.  The  former  component  is  H,  the  latter 
Hoc  sin  <pdt  (Ex.  12),  and  the  two  are  at  right  angles.  Hence  the  momentum 
axis  OB  turns  towards  the  centrifugal  couple  axis  OB'  with  an  angular  velocity 
to  sin  <p,  which  is  equal  and  opposite  to  the  angular  velocity  of  the  body  round 
OQ,  the  perpendicular  to  the  momentum  axis  and  the  centrifugal  couple  axis. 


374 


Kinetics  of  a  Rigid  Body. 


16.  The  axis  of  the  centrifugal  couple,  regarded  as  a  radius  vector  of  the 
ellipsoid  of  gyration,  describes  areas  proportional  to  the  time  in  the  invariable 
plane  of  principal  moments. 


Let  OR  be  the  momentum  axis  and  OR'  the  centrifugal  couple  axis  at  any 
instant.  Let  the  lengths  of  OR  and  OR',  regarded  as  radii  vectores  of  the  ellipsoid 
of  gyration,  be  R  and  r.  Describe  a  sphere  with  radius  R  round  0  as  centre, 
and  "let  OP  and  OP'  be  the  positions  of  the  actual  axis  of  rotation  and  of  the 
axis  of  rotation  due  to  centrifugal  forces  at  the  instant.  Let #  the  body  be 
rotating  clockwise  round  OP.  At  the  end  of  the  time  <ft  the  instantaneous 
axis  of  rotation  will  have  turned  towards  OP',  into  the  position  01. 

In  the  figure  the  line  OR  and  the  plane  at  right  angles  to  it  are  fixed  in 
space,  whilst  OP  and  01  are  consecutive  positions  in  space  of  the  instantaneous 
axis  of  rotation.  The  position  of  the  centrifugal  couple  axis  at  the  end  of  the 
time  dt  is  at  right  angles  to  tbe  plane  RIQ'.  If  dv  be  the  angle  described  by 
this  axis  in  the  time  dt,  dv  =  QOQ' .  If  w'  be  the  angular  velocity  produced  in 
the  time  dt  by  the  centrifugal  couple,  we  have  (Ex.  12  and  13,  and  Ex.  6, 
Art.  264) 

moo'  p'r  =  niocrpR  sin  (pdt ; 


whence 


uRp  sin  <j>  dt      go        sin  PI 


rp 


dv 


sin  P'l 

sin  RP 


sin  PI    sin  IR 
sin  IR  '  sin  PI 


dv  sin  <p 
sin  P'PQ  '  sin  P'P       cos  tf>' 


but 


whence,  finally, 


r  cos  (p'}    and    p  =  R  cos  <\> ; 
r2  dv  =  ooR?  cos  <p  dt ; 


and  as  R  and  «  cos  <p  are  each  constant,  the  theorem  is  proved.  It  is  to  be 
observed  that  it  follows  from  the  equations  of  Ex.  13  above,  and  Ex.  5,  Art.  264, 
that  the  angles  ROP  and  R'OP'  are  each  acute. 


Examples. 


375 


Another  mode  of  proving  the  theorem  contained  in  this  example  is  as 
follows  : — 

,    .    „  W       dv  sin  <p 

As  before,  —  =  ■£■  ; 

to         cos  <p 


therefore 


dt 


dv    to  sin  (p 
dt     cos*' 


Again,  from  Exs.  12  and  13,  we  readily  see  that 
to'       S  tan  $  cos  tf>'        S  tan  <p 


dt  mp"1  mr~  cos  (p 

Hence,  equating  these  values  of  — ,  we  get 


■,  since  p'  =  r  cos  <p' 


r2  —  = =  —    by   9),  Art.  264. 

dt       into  cos  <p       m 

17.  To  determine  the  position  of  the  body  in  space  at  any  time. 

The  line  in  the  body  which  at  a  given  time  coincides  with  the  momentum 
axis  is  known  from  Art.  269.  If,  then,  we  make  this  line  of  the  body 
coincide  with  the  momentum  axis  (whose  position  in  space  remains  unaltered), 
and  then  turn  the  body  through  the  proper  angle  round  this  axis,  the  position 
of  the  body  in  space  is  determined. 

To  effect  the  latter  part  of  the  determination,  we  consider  the  position  in 
space  of  the  line  OQ,  which  is  at  right  angles  to  the  momentum  axis  and  the 
axis  of  the  centrifugal  couple. 

The  momentum  axis  OR  describes  a  cone  C  in  the  body,  and  it  is  easily  seen 
that  OQ  describes  the  reciprocal  cone  C.  The  position  of  the  momentum  axis 
in  the  body  being  known,  so  likewise  is  the  position  of  OQ,  the  corresponding 
edge  of  C;  and  if  we  can  determine  the  position  of  the  latter  in  space,  the 


problem  is  solved.  The  instantaneous  axis  of  rotation  OP  lies  in  the  plane  of 
OR  and  OQ  ;  and  the  angular  velocity  to  round  <9Pis  equivalent  to  w  cos  <p  round 
OR,  and  co  sine/)  round  OQ.  If,  then,  we  suppose  the  cone  C  rigidly  connected 
with  the  body,  the  whole  motion  consists  of  the  rolling  and  sliding  of  the  cone 


376 


Kinetics  of  a  Rigid  Body. 


C  on  the  plane  of  principal  moments.  The  angular  velocity  of  sliding  is 
io  cos  <p  ;  and  if  de  be  the  angle  between  two  consecutive  edges  of  the  cone  C,  the 

velocity  with  which  OQ  turns  in  consequence  of  the  rolling  is  — .     Hence,  if 

the  cone  C  be  on  the  same  side  of  the  plane  of  principal  moments  as  OP,  the 
whole  angular  velocity  of  the  edge  round  OR  is 

ch 
(O  COSrf)  — — . 
r      dt 

Thus,  if  the  rotation  round  OP  be  counter-clockwise,  a>  cos  <p  will  be  from  R'to 
Q,  whilst  the  rolling  round  OQ  will  bring  an  edge  into  contact  with  the  plane 
of  principal  moments  which  is  nearer  to  OH',  and  this  will  impart  an  angular 

velocity to  the  edge  of  6"  which  is  in  the  plane  of  principal  moments.   We 


can  arrive  at  the  same  result  in  another  way,  by  considering  the  motion  of  the 
cone  C  regarded  as  rigidly  attached  to  the  body.  The  rotation  round  OQ 
(supposed  counter-clockwise)  brings  the  line  OS\  consecutive  to  OR  into  the 
position  OR,  and  moves  the  next  consecutive  tangent  plane  of  C  in  a  direction 
parallel  to  OR' .  The  next  rotation  is  effected  round  a  line  at  right  angles  to 
OSi  £2,  which  is  therefore  nearer  than  OQ  to  OR' :  thus  the  motion  of  OQ,  in 
consequence  of  a  counter-clockwise  rotation  round  OQ,  is  clockwise.  On  the 
other  hand,  the  motion  of  OQ,  in  consequence  of  a  counter-clockwise  rotation 
round  OR,  is  likewise  counter-clockwise. 

Hence,  on  the  whole,  if  v  be  the  angle  described  by  the  line  at  right  angles 
to  the  momentum  axis  and  the  centrifugal  couple  axis  in  the  fixed  plane, 


v  =  j  w  cos  (p  dt  —  I  —  dt. 


Examples,  377 

Since  de  is  the  angle  between  two  consecutive  edges  of  the  cone  C", 

P' 

if  s  be  the  arc  of  the  spherical  conic  in  which  the  cone  C  meets  the  sphere  of 
radius  P  ;  whence,  finally, 

s 
v  =  cut  cos  0  — — . 
P 

If  the  cone  C  be  on  the  opposite  side  of  the  plane  of  principal  moments 
from  OP  and  OP,  or,  in  other  words,  if  the  curvature  of  the  cone  0  turned 
towards  OP  be  convex  instead  of  concave,  the  two  parts  of  the  motion  of  OQ 
have  the  same  sign,  and 

s 
v  =  cot  coscp  -f  — . 
P 

The  former  case  occurs  when  P  is  less  than  the  mean  axis  of  the  ellipsoid 
of  gyration  ;  the  latter,  -when  it  is  greater.  In  either  case  s  is  determined  by 
knowing  the  position  of  OQ  in  the  cone  C . 

In  order  to  facilitate  the  drawing  of  the  figures,  the  rotation  is  in  Ex.  16 
supposed  to  be  clockwise,  but  in  the  present  example  counter-clockwise. 

It  is  to  be  observed  also  that  in  the  figure  of  the  present  Example  the  angle 
QOQ'  represents  only  part  of  the  motion  of  OQ,  viz.  de,  while  in  the  figure  of 
Ex.  16  it  represents  the  whole  motion  dv. 

Examples  12  to  16  are  due  to  Mac  Cullagh. 

18.  The  normals  to  the  cone  described  by  the  instantaneous  axis  of  rotation 
intersect  the  ellipsoid  of  gyration  in  a  line  of  curvature. 

The  normal  to  the  plane  of  OP  and  OP'  (figure  of  Ex.  16)  is  conjugate  to 
OP  and  OP',  and  hence  passes  through  the  intersection  of  the  ellipsoid  of  gyra- 
tion with  the  confocal  a2  -  P2.     This  result  follows  also  from  Ex.  5. 

19.  Show  that,  if  v  and  <p  have  the  same  meaning  as  in  Ex.  16, 

dv       S       (AS-S2)(BS-R2)(CS- H2) 
Tt=H  + ABCHS2 COt"  *' 

Substituting,  by  (9)  and  Ex.  8,  Art.  264, 

O  T7"2 

o)  cos  <j>  for  — ,  P-  for  — ,  and  mar  for  A,  mP  for  B.  mc-  for  C, 
R  mS 

the  equation  given  above  is  reduced  to 

—  =  co  cos  <b    1  +  ^ M  ,)70  „ cot2  <b   . 

at  (  a1  b-c~  ) 

Now,  by  Ex.  16,  —  =  u  cos  <b  —  ; 

dt  r- 

P2      ,       (a2-P2)(b2-P2)(c2-P2)        _ 
also  — -  =  1  + „ .,  0 cot2  (p, 


a2  b2  c2 


as  may  be  thus  proved. 


378  Kinetics  of  a  Rigid  Bodij. 

The  axes  of  a  central  section  of  the  quadric  a,  are  parallel  to  the  normals- 
to  the  two  confocals  through  the  extremities  of  the  semi-diameter  B  conjugate 
to  the  section,  and  (a',  a"  being  the  semi-axes  of  the  confocals)  are  given  by 
the  equations 

R?  -  a2  —  a'2,     r2  =  a2  -  a"2 

(Salmon's  Geometry  of  Three  Dimensions,  §  164).  Moreover,  if  /be  the  axis 
normal  to  a"  of  the  section  conjugate  to  D  in  the  quadric  a',  the  direction  of  r' 
coincides  with  that  of  r,  and  the  magnitude  of  r'  is  given  by  the  equation 

r'2  =  a'2  -  a"2  =  r2  —  R2. 

Again,  if/  be  the  perpendicular  on  the  tangent  plane  to  a',  which  is  parallel  to 
the  plane  of  D  and  r  or  r, 

p'z  =  a'2  cos-  a  +  b'2  cos2£  +  c'2  cos2  y  =  p2  -  It2, 

since  a'2  =  a2  -  R2,     V2  =  b2  -  R2,     c'2  =  c2  -  R2. 

Hence,  if  ^  be  the  angle  between  D  and  the  direction  of  r  or  r',  we  have,  from 

the  quadric  a, 

_„         a2  b2  c2 
D3  = 

r2p2  sin2i|/ 

and,  from  the  quadric  a', 

a'2  b'2  c'2 
D2  = 


r'2  p'2  sin2v|/ 
and  therefore 

#v*  j# -*){»-&){*-&).  but  _£_-00f^. 

r2p2  (r2  -  R-)(p2  -  Rl)        '  Rz  - p2 

„     „  R2      «      (a2  -  R2)  (b2  -  R2)  (e  -  2t2) 

whence,  finally,         —  -  1  =  K- ,  „  ,   - '  cot~  *• 

J  r2  a2b-c2 

The  expression  given  in  this  example  for  —  is  due  to  Poinsot. 

20.  Determine  the  differential  equation  of  the  herpolhocle  {see  Ex.  7). 

If  p  and  v  be  the  polar  coordinates  of  the  point  of  contact  of  the  momenta! 
ellipsoid  with  the  invariable  plane,  the  origin  being  the  foot  of  the  perpendicular 
on  it  from  the  fixed  point,  we  have 


\K  -JKS  ..   L   naA.         ,  dv       S   (  p2\ 

where      r  =  ^—u>,    p  =  _  (Art.  2o4),  and  —  =  —  ^1  +/*  ^  J 


where  M  -  ^^ 


(AS_-IP)(BS-B*){CS-IP)    (Ex  19) 


Impact. 


379 


If  we  express  co  in  terms  of  p,  on  substituting  in  equation  (24),  Art.  269, 
we  get 


i-s-^Ji[-4^^][-^^] 


A3 


7,(P2+    P") 


]l- 


274.  Impact, — When  two  smooth  bodies  moving  in 
any  way  collide,  the  results  of  the  impact  are  obtained  in 
a  manner  precisely  similar  to  that  employed  in  Article  243. 

When  the  motion  is  wholly  unrestricted  there  are  thirteen 
unknown  quantities  and  thirteen  equations. 

If  A,  ju,  v  be  the  angles  made  by  the  common  normal  at  the 
point  of  contact  with  axes  fixed  in  space ;  R  the  whole  impulse 
of  the  mutual  normal  action  during  the  first  period  of  impact ; 
p  and  p  the  perpendiculars  on  its  line  of  action  from  the 
centres  of  inertia  of  the  two  bodies ;  a,  /3,  y,  a,  /3',  y  the 
angles  made  with  the  principal  axes  of  the  bodies  by  the  axes 
of  the  couples  produced  by  R  round  these  points  ;  twelve  of 
the  equations  mentioned  above  are 


SDct*  =  R  cos  A, 
9Jte    =    R  cos  p., 

yjliv  =  R  cos  V, 
Wu '  =  -  R  cos  A, 
Wv  =-Rcosp, 
Wl'tv'  =  -  R  cos  v, 


Azji  =  Rp  cos  a 
Btu2  =  Rp  cos  j3 
C-&z   =  Rp  cos  7 
A'zs{  =  -  Rp  cos  a 
JBV  =  -ify,cos/3/ 

O'W  =  -  ify>'  COS  y' 


>,     (46) 


where  ««,  &c,  are  the  changes  of  the  components  of  the  velo- 
city of  the  centre  of  inertia  of  the  first  body,  parallel  to  axes 
fixed  in  space,  produced  during  the  first  period  cf  impact ;  _  zsx+ 
&c,  the  changes  of  the  angular  velocities  round  the  principal 
axes  through  the  centre  of  inertia  produced  during  the  same 
period  ;  and  u,  &c,  have  similar  significations  for  the  second 
hody. 

At  the  end  of  the  first  period  the  actual  components  of 
the  velocity  of  the  centre  of  inertia  of  the  first  body  are 
u  +   k0,  &c.,   where  4  represents   the   component    of   this 


380  Kinetics  of  a  Rigid  Body. 

velocity  immediately  before  the  impact.  In  like  manner,  tsx  +  Q,x 
is  the  actual  angular  velocity  round  the  first  principal  axis. 
"We  can  then  write  down,  in  terms  of  t#  +  x0i-  tsl  +  £2b  &c, 
the  relative  normal  velocity  of  the  points  of  the  two  bodies 
which  are  in  contact.  Equating  this  relative  normal  velocity 
to  zero  gives  a  thirteenth  equation  ;  so  that  te ,  zsi9  &c,  become 
completely  known. 

If  x  be  the  component  of  the  final  velocity  of  the 
centre  of  inertia  of  the  first  body  at  the  end  of  the  second 
period  of  impact,  and  wi  the  final  angular  velocity  round 
the  first  principal  axis,  &c,  the  values  of  the  velocities  at 
the  end  of  the  impact  can  now  be  determined,  by  aid  of  the 
following  equations — 

x  -  x0  =  (1  +  e)  t«,     on  -  Qi  =  (1  +  e)  zju  &c,  ) 

.    (47) 

x'  -  x0'=  (1  +  e)  u,     u){-  Q,i-  (1  +  e)  ot/,  &c.   ) 

Since  the  positions  of  the  two  bodies  are  not  sensibly 
altered  during  the  whole  period  of  impact,  it  is  to  be  ob- 
served that  throughout  this  period  any  lines  fixed  in  either 
body  coincide  with  lines  fixed  in  space. 

275.  Impulsive  Friction. — When  collision  takes  place 
between  two  rough  surfaces  we  can  investigate  the  motion 
according  to  the  principles  laid  down  in  Article  247. 

The  elementary  impulse  dF  of  friction,  at  each  instant  of 
the  impact,  is  to  be  resolved  into  two  components,  dP  and  dQ, 
along  two  tangents  through  the  point  of  contact  at  right 
angles  to  each  other.  At  any  instant  during  the  impact,  P 
represents  the  entire  impulse  in  a  given  direction  due  to  the 
action  of  friction  up  to  that  instant.  A  similar  remark 
applies  to  Q,  and  R  is  the  corresponding  impulse  due  to 
the  normal  reaction. 

If  at  any  instant  during  the  impact  u,  v,  w  be  the  com- 
ponents, along  the  two  tangents  and  the  normal,  of  the 
relative  tangential  and  normal  velocities  of  the  points  of  the 
two  surfaces  which  are  in  contact,  u,  v,  w  can  be  expressed  in 
terms  of  the  velocities  of  the  two  centres  of  inertia  and  of 
the  angular  velocities  of  the  bodies  at  that  instant ;  they  are 


Collision  of  Rough  Spheres. 


381 


therefore  linear  functions  of  P,  Q,  R.  If  slipping  take  place 
its  direction  coincides  with  that  of  the  elementary  impulse  of 
friction,  and  therefore 

dQ  =  v  '    als°    ^  ^dF*  +  d®)  =  fxdR' 
Initially  R  is  zero,  and  therefore  so  likewise  are  P  and  Q, 
except  the  colliding  surfaces  be  perfectly   rough.      When 
R  =  Rl9  at  the  end  of  the  first  period  of  impact,  w  =  0 ;  and 
if  R2  be  the  value  of  R  at  the  end  of  the  whole  impact, 

P2  =  (1  +  e)  R,. 

If  the  surfaces  which  collide  be  perfectly  rough.,  the  equa- 
tions u  =  0,  v  =  0,  w  =  0  enable  us  to  determine  Pl9  Ql9  R{. 
Knowing  the  value  of  Rz  we  can  find  P2  and  Q2  from  the 
equations  u  =  0,  v  =  0,  which  hold  good  throughout  the 
impact. 

If  the  bodies  slip  on  each  other  in  the  same  direction 
during  the  whole  of  the  impact,  the  direction  of  clF  is  con- 
stant, and  we  may  take  clQ  =  0,  clP  =  /mdR.  Hence  P,  =/uRu 
Qi  =  0  :  these  equations,  with  w  =  0,  determine  Rx ;  then 

P2  =  fi  (1  +  e)  Rl 

276.  Collision  of  Rough  Spheres. — If  a  homogeneous 
sphere  impinge  against  a  fixed  surface,  or  two  homogeneous 
spheres  collide  with  each  other,  by  taking  as  axes  of  <r,  y,  z 
parallels  to  two  tangents  and  the  normal  at  the  point  of  con- 


tact, 

at  any 

instant  dm 

ing 

the 

impact,  we 

have 

x  = 

x0  - 

P 

y  = 

$0 

Q      -• 

^0   " 

R 

x  = 

x£  + 

P 

y  = 

9i 

-4- g   - 

-  f 

-  20 

R 

+  W 

CU] 

=  a, 

aQ 
+  J3 

U> 

2  == 

OJ-i 

=  Q3, 

*Jn+—,    ,        Wo,  =  ii2 — -, 


(1)3=  Q/, 


J 


>  (48> 


where  0,  Wl,  and  Pare  the  radius,  mass,  and  moment  of  inertia 


382  Kinetics  of  a  Rigid  Body. 

round  its  diameter,  of  the  first  sphere  :  x,  &c,  the  compo- 
nents of  the  velocity  of  its  centre,  wi,  &c,  the  components 
of  its  angular  velocity :  x0,  &c,  and  Oi,  &c,  the  values  of 
these  components  immediately  before  the  impact :  and  a',  2D?', 
&c,  have  similar  significations  for  the  second  sphere. 

At  the  same  instant  the  velocities  of  the  points  of  the 
spheres  which  are  in  contact  are  given  by  the  equations 


TJOJ3 


+  Z(t)2,      X*  =  X   -  ri'to-/  +  ?Vs',  &C., 


where  £,  ri,  Z ',  £',  v',  Z',  are  the  coordinates  of  the  point  of 
contact  relative  to  the  centres  of  the  two  spheres ;  and  the 
relative  velocities  u,  i\  to  are  then  determined  by  the  equa- 
tions u  —  x—  X,  &c. 

Hence,  since  £  =  0,  r\  =  0,  £  =  a,  £'  =  0,  r{  =  0,  Z'  =  -  a', 
substituting  for  x9  tou  &c.,  their  values  given  by  (48),  we  have 

,     /     (1         \        a2      d2\„ 
u  =  x0  -  x0  +  aQ2  +  a  £22  - 1  ^  +  —-,  +  —  +  j,f, 

'        '  /       „         ;/1,    /l        1        a2      d2\  _ 
v  =  V*  -  Vo  -  aQl  -  a  Ql  -( ^  +  ^p  +  j  +  y)  & 

that  is         m  =  Mo  -  /P,     #  =  «?o  -  JQ,     w  =  Wo  -  nR, 
where  /  and  n  are  constants. 

Hence     die  =  -  IdP,     dv  =  -  IdQ,     —  =  -jp.  ; 

dv      dQ 

but,  if  there  be  slipping,  we  have,  Art.  275, 

—  =  -  ;  therefore    —  =  -,  and  accordingly  -  is  constant, 
dQ      v  dv      v  v 

or  the  direction  of  slipping  is  invariable  throughout  the  impact. 
Moreover,  if  either  u  or  v  vanish  so  must  the  other.  All. 
slipping  then  ceases,  and  cannot  recommence,  as  u  and  v 
are  independent  of  R,  and  friction  cannot  initiate  slipping. 
Since,  in  the  present  case,  Pi  is  independent  of  P  and  Q,  if 
there  be  no  slipping  at  the  end  of  the  impact  the  result  is  the 
same  as  if  there  had  been  no  slipping  at  all. 


Ejcamjrfes. 


383 


Hence,  in  all  cases,  either  the  impulse  of  friction  is  a 
maximum,  and  the  direction  of  slipping  the  same  throughout 
the  impact,  or  else  the  surfaces  may  be  regarded  as  perfectly 
rough. 

If  the  problem  be  solved  on  the  latter  hypothesis,  and  the 
value  resulting  loi*/(P?  +  Q22)  does  not  exceed  fi(l  +  e)  B„ 
the  solution  is  correct.  If  </(P*  +  Q22)  be  greater  than 
ju(l  +  e)  Bi,  slipping  takes  place  in  the  same  direction  through- 
out the  impact. 

Examples. 

1.  A  sphere,  having  no  original  velocity  of  rotation,  impinges  successively 
against  two  perfectly  rough  vertical  walls  at  right  angles  to  each  other,  the 
points  of  impact  heing  so  near  the  intersection  of  the  walls  that  the  action  of 
gravity  hetween  the  two  impacts  may  he  neglected ;  determine  the  magnitude 
and  direction  of  the  velocity  of  the  centre  of  the  sphere  after  the  second  impact. 

Take  as  axes  of  f,  77,  (,  three  lines  through  nT 

the  centre  of  the  sphere  parallel  to  the  inter- 
sections of  the  walls  with  the  plane  of  the 
horizon  and  with  each  other.  Let  the  com- 
ponents of  the  velocity  of  the  sphere  before 
the  first  impact  he  Vi,  V2,  V3,  and  let  the 
sphere  impinge  first  against  the  plane  YO'Z'. 

At  the  first  impact  the  coordinates  |,  77,  £ 
of  the  point  of  contact  are  given  by  the  equa- 
tions 

1  =  0,     n  =  a,     £=0. 

Hence,  Art.  255,  (3),  °  X 

x  =  x-  awz,     y-y,    z  =  z  +  «»i, 

where  x,  y,  z  are  the  coordinates  of  the  point  of  contact  referred  to  axes  meeting 
at  0. 

Again,  X,  Z  and  R  heing  the  entire  impulses  due  to  friction  and  to  the 
normal  reaction  up  to  the  end  of  the  first  period  of  the  collision,  we  have 


z 

T 

/ 

/ 

/ 

mx  =  in  V\  +  X,     my  =  m  Vi  +  It,     tnz  =  m  Vz  +  Z, 


ima'wi 


aZ,     toz  =  0,     %ma2a>z  =  -  aX,     x=0,     y  =  0, 


0. 


Hence 


=fPij     y=0,     2=7^3,     «wi=-fr3,     a  o>2=0,     flw3  =  fFi 


At  the  end  of  the  second,  period  of  the  collision  five  of  the  six  velocities 
above  remain  unaltered,  but  y  becomes  —  eVt. 

At  the  second  impact  the  coordinates  of  the  point  of  contact  are 

S  =  «,     T7  =  0,     C  =  o. 


Hence, 


=  y  +  ao>3,     z  =  z  —  aw2. 


384  Kinetics  of  a  Rigid  Body. 

Also,  as  has  teen  proved  above, 

Proceeding  as  before,  since  y  =  0,  z  =  0,  we  obtain 

awi  =  -  f  Ts,     «o>2  =  (y)2  T3,     aW3  =  f e  Fa  +  f .  f  Fi, 

J— *Pi,  ?=- *r»-*.*Fi,  MfpFs. 

2.  In  the  last  example,  if  the  walls  be  not  perfectly  rough,   determine  the 
final  velocities  of  translation  and  rotation. 

We  first  treat  the  question  as  before,  and  obtain,  as  in  the  last  example,  at 
the  first  impact, 

1  (X2  +^2)=-A-(Fi2  +  F32);  if  then  M(l  +^)r2>f  V(Fi2+  F32) 

we  may  assume  there  is  no  slipping ;  but  if  this  condition  is  not  fulfilled  slipping 
takes  place,  and  the  maximum  amount  of  friction  is  exerted.  In  this  case  at 
the  end  of  the  first  impact, 

x=  V\—  fi{l  +  e)Vt  cos  a,     y  =  -eV^     z—  F3  —  fi{l  +  e)  F2sinff, 

«o>i  =  — |/t(l  +e)Vz  sin  a,     co2  =  0,     «w3  =  f  /x(l  +e)  F2cos  a, 

where  tan  a  =  — . 
'i 
The  values  of  x,  &c,  »i,  &c,  at  the  end  of  the  first  impact  are  the  values 
of#o,  &c,  Xii,  &c.,  at  the  second  impact.     If  slipping  takes  place  during  the 
whole  of  the  second  impact,  we  have  finally 

z  =  -e{Vi  -  fi{l  +  e)VzCOSa}, 

y-  =  _  gf2  -  ^(1  +  e){  Vi  -  fi(l  +  e)V2  cos  a]  cosj3, 

|  =  r3  -  /*(1  +  e) {  r2  sin  a  +  Vi  sin  /3  -  p. (1  +  *)  V%  cos  a  sin  £}, 

«Wi  =  0,     «W2  =  f/*(l  +  «){Fi-/t(l  +  ^)F2  cos  a}  sinjS, 

<?OT3  =  -|/x(l  +  e){Vi  -  /x(l  +  e)F2cosa}  cos£, 
flwi  =  -  f  fi{l  +  e)  F2  sina, 
oa>2  =  f  /*(1  +  «){Fi-/i(l  +  «)F2  cos  a}  sin  0, 
«fi>3  =  | /*(1  +  e){F2cosa[l  +/*(1  +  e)cos/3]-  Ficos/3}, 
2{  r3-/*(l  +^)F2sina} 


where  tan  /3  = 


5/a(1  +  <?)F2  cosa-  2eF% 


Motion  consisting  of  Successive  Rotations. 


385 


277.  Equations  of  motion  referred  to  Body-Axes. 

— If  u,  v9  w  be  the  components  of  the  velocity  of  the  centre 
of  inertia  at  any  instant  in  the  directions  of  the  principal 
axes  of  the  body  at  that  point,  the  accelerations  of  the  centre 
of  inertia  in  these  directions  are,  by  Art.  257, 


clu 

—  -  Vd)z  +  U'W2, 

at 


&c. 


whence,  if  SZ,  S7,  2Z  be  the  sums  of  the  components  of  the 
applied  forces  at  any  instant,  parallel  to  the  principal  axes 
through  the  centre  of  inertia,  its  equations  of  motion  are 


cm  \du 

VJC     —  -  0W3+  Wbh\ 

\dt 


2Z 


m^-mo^uco3\=^Y 


(49) 


278.  Motion  consisting  of  Successive  Rotations. — 

If  the  whole  motion  of  a  body  consist  of  successive  rotations 
(not  necessarily  executed  round  lines  passing  through  the 
same  point),  the  vis  viva  at  any  time  is  Jw2,  where  i"  is 
the  moment  of  inertia,  and  w  the  angular  velocity  round  the 
instantaneous  axis ;  hence 

Jcu2  =  22  \(Xdx  +  Yd//  +  Zdz)  +  const,  ; 


therefore 


d  ,r  ox      ^  /  -xrdx     __f/y     „dz\ 


Let  p  be  the  perpendicular  on  the  instantaneous  axis  from 
the  point  %,  //,  z ;  and  let  the  direction  of  the  motion  of  this 
point  make  angles  a,  j3,  y  with  the  axes  ;  then 


dx 
dt 


pu)  cos  a, 


dy 
dt 


pu)  cos  /3, 

2C 


dt 


pu)  cos  y ; 


386  Kinetics  of  a  Rigid  Body. 

whence   —  (iw2)  =  2u)^p  (X  cos  a  +  Y  cos  j3  +  Z  cos  y  ; 

but  (X  cos  a  +  Y  cos  j3  +  Z  cos  7)^ 

is  the  moment  of  the  force  applied  at  the  point  x,  y,  z  round 
the  instantaneous  axis.  Hence,  if  J  be  the  moment  of  the 
entire  system  of  applied  forces  round  the  instantaneous  axis, 
we  get 

or  -?-(I<S)  =  2J.  (50) 

(1)  at 

If  the  body  be  such  that  the  moments  of  inertia  round 
the  different  instantaneous  axes  are  equal,  this  equation  takes 
the  simple  form 


Examples. 

1.  A  homogeneous  sphere,  having  an  initial  angular  velocity  round  a  hori- 
zontal axis,  is  projected  along  a  rough  horizontal  plane :  determine  the  motion, 
neglecting  the  couple  of  rolling  friction.   (Jellett,  Theory  of  Friction,  Chap.  V.). 

The  axes  heing  three  mutually  perpendicular  lines  through  the  centre, 
whose  directions  are  fixed  in  space,  equations  (17)  of  Art.  267  become  for  a 
homogeneous  sphere  whose  radius  is  r, 

at  at  at 

where  Z,  M,  N  are  the  moments  of  the  forces  round  the  axes. 

Let  X  and  Y  be  the  components  of  friction  along  two  horizontal  axes,  x  and  y 
the  coordinates  of  the  centre  of  the  sphere,  u  and  v  the  components  of  the  velo- 
city of  its  point  of  contact  with  the  rough  plane  ;  then,  by  Art.  255, 

dx  dy 

u  = rw2j     v  =  - — h  rw\. 

dt  at 


The  equations  of  motion  are 

dt2  dt2  dt  dt 


Examples.  387 


Combining  the  first  and  last  of  these,  we  have 
du      drx         dw2       » X 


—  r 


~  —  z 


dt       dt2  dt       zm 

In  like  manner 

dv      „  Y         ,  du      X 

T-  =  t™j     whence  — =  -. 

Now  if  there  be  slipping, 

^  CAV,  U  TT  r**  v  X  u 

whence  -  =  constant  =  cot  o  =  — -.   where   Vu  Vi,  fli,  n2  are  the  initial 

v  V-z  +  rD.1 

values  of  —  ,     — -,     a>i,  and  o>2.     Hence 
dt       dt 

X  =  —  yu9%  cos  a,      Y  =  -  ^9%  sin  a. 

These  are  the  components  of  a  constant  force  in  a  fixed  direction.  Hence  in 
general  the  centre  of  the  sphere  describes  a  parabola.  If,  however,  the  initial 
axis  of  rotation  be  perpendicular  to  the  direction  of  the  initial  motion  of  the 
centre,  i.e.  if  V\Cl\  +  VzQz—O,  the  centre  of  the  sphere  continues  to  move  in 
the  direction  of  its  initial  motion. 

Substituting  the  values  of  X  and  Y  in  the  equations  of  motion,  we  find  that 

2  ( Vi  -  ra-) 

slipping  ceases  along  the  axis  of  x  when  t  =    „ - ,  and  along  the  axis  of  >/ 

**    °  7 /iff  cos  a    '  °  J 

,  2(V2+rQi)    ..Vz+rQi       Fi  -  ril3  . 

when  t  =  — : ;  but  — : = ,  hence,  slipping  along  each  axis 

7 /x.ff  sin  a  sin «  cos  a  irr     o         o 

ceases  at  the  same  time,  to,  where 

t     gV{(ri-m2)2+(r2+rfli)a} 

to  =  t ' 

H-ff 

After  pure  rolling  begins  it  will  continue,  since  the  values  which  X  and  Y 
must  take  in  order  to  maintain  it  are  zero  ;  the  components  of  the  velocity  of 
the  centre  are  then  given  by  the  equations 

dx  _  5  Vi  +  2>-n2       dy  _  5  Vi-  2rHi 

¥ "  7         '     <fc  "  7        ' 

and  the  remainder  of  the  path  is  a  straight  line.  If  /3  be  the  angle  which  the 
final  line  of  motion  of  the  ball  makes  with  the  axis  of  x, 

5F2-2rfl, 

tan£ 


5  Vi  +  2rn2 


The  result  here  obtained,  that  the  centre  of  the  sphere  may  describe  a 
parabola,  enables  us  to  explain  a  well-known  phenomenon  in  billiards,  which 
may  be  stated  as  follows  : — 

The  angle  at  which  the  striker' shall  goes  off  the  ball  aimed  at,  in  order  to  make 
a  cannon,  seems  to  be  less  according  as  the  distance  of  the  third  ball  is  greater. 

2  C  2 


388 


Kinetics  of  a  Rigid  Body. 


Let  A  be  the  striker's  ball,  B  the  ball  first  struck  by  A.     If  A  be  struckin 


the  ordinary  way,  without  side,  it  will  have 
round  a  horizontal  axis  CR  at  right  angles  to 
SC,  the  line  of  original  motion.  This  rotation 
will  continue  round  the  same  axis  after  im- 
pact. Suppose  the  motion  of  translation  after 
impact  to  be  in  the  direction  CT,  then  CR  not 
coinciding  with  CR',  the  horizontal  line  per- 
pendicular to  CT,  the  path  described  by  the 
ball  is  the  parabolic  arc  CPP' .  Hence,  if  P' 
be  more  remote  than  P,  PCS'  is  greater  than 
P'CS'. 

It  is  well  known  that  a  skilful  billiard 
player  can  make  a  ball  describe  a  very  marked 
curve.  This  is  done  by  an  impulse  having 
a  vertical  component  which  imparts  a  rotation 
round  the  line  of  original  translation  of  the 
centre. 

If  the  original  impulse  be  horizontal  it 
produces  no  moment  round  a  line  through 
the  centre  parallel  to  itself,  and  this  latter 
being  the  original  line  of  translation  of  the 
centre,  there  can  be  no  rotation  round  it ; 
hence  in  this  case  the  ball  must  move  in  a 


when  it  strikes  B,  a  rotation 


straight  line. 


2.  A  sphere  rolls  along  a  rough  horizontal  plane.  Taking  into  account  the 
couple  of  rolling  friction  determine  the  forces  brought  into  play  and  the  path 
described,  the  motion  being  pure  rolling. 

The  equations  of  motion  are 


m  d2x      , 


m 


a-J=T, 


f  Wlr- 


c/i 


■T-ffDlff 


wh( 


=  Vc 


with  the  conditions 


dx  dy 

—  -  ru2  =  0,    — -  +  rux  =  0, 
dt  dt 


whence 


X 


^-^A     Y^f-Wg^ 


-  o;i 

G>2 


Hence  the  path,  is  a  straight  line.     Multiplying  the  third  equation  of  motion  by 
co  i,  the  fourth  by  «2,  and  adding,  we  have 

dt 


whence 

n  being  the  initial  angular  velocity 


ra  =  -  |  fjL  t  +  ra, 


Examples.  389 

7r2a 
The  sphere  will  come  to  rest  when  t  =  — — . 

¥</ 
Again,  multiplying  the  third  equation  of  motion  by  W2,  the  fourth  by  «i,  and 

co\ 
subtracting,  we  have  a>2^wi  —  widw2  =  0.     Hence  —  =  constant  =— tan  a,  where 

too 

a  is  the  angle  which  the  path  makes  with  the  axis  of  x. 

3.  A  sphere  is  projected  obliquely  down  a  rough  inclined  plane,  the  motion 
being  pure  rolling  ;  determine  the  friction  brought  into  play,  and  the  path,  neg- 
lecting the  couple  of  rolling  friction. 

Take  as  axis  of  x  the  intersection  of  the  inclined  plane  with  'a  vertical 
plane  at  right  angles  thereto. 

The  equations  of  motion  are 

dt-  dt*  dt  at 

...  ..'.  dx  dy 

with  the  conditions  —  —  ra>2  =  0,     —  4-  ra\  =  0  ; 

dt  at 

whence  Y=  0,     X  =  -  f  Tig  sini. 

The  whole  force,  therefore,  is  f  Tig  sin  i  parallel  to  the  axis  of  x,  and  the 
centre  of  the  sphere  being  acted  on  by  a  constant  fotce  parallel  to  a  fixed  direc- 
tion, describes  a  parabola.     Also,  since 

dx  dy 

dt  dt 

the  instantaneous  axis  of  rotation  is  at  right  angles  to  the  tangent  to  the  path 
of  the  point  of  contact  on  the  inclined  plane.  This  is  otherwise  immediately 
obvious,  since  the  motion  is  pure  rolling. 

279.  Equations  of  Motion  of  a  Solid  of  Revolu- 
tion.— If  one  point  0  of  a  rigid  body  be  fixed  in  space,  and 
two  of  the  principal  moments  of  inertia  at  the  point  be  equal, 
the  equations  of  motion  of  the  body  can  be  expressed  in  a 
comparatively  simple  form. 

Let  00  (Art.  258)  be  the  axis  of  revolution  of  the 
momental  ellipsoid  of  the  body,  and  A  and  0  its  principal 
moments  of  inertia  at  0,  then,  by  considering  the  motion  of 
a  point  situated  on  OO,  it  is  plain  that  the  angular  velocity 
of  the  body  round  an  axis  OS  perpendicular  to  OO  in  the 
plane  ZOO  is  \p  sin  9,  and  the  moment  of  momentum  round 
OS  is  therefore  A\p  sin  9.     Hence 

Ez  =  Axp  sin2  9  +  Cw2  cos  9.  (52) 


390  Kinetics  of  a  Rigid  Body. 

Again,  the  angular  velocity  of  the  body  round  OE  perpen- 
dicular to  the  plane  ZOC  is  9,  and  therefore  we  have 

2T  =  A  (^2sin2  9  +  92)  +  CW.  (53) 

If  Gx,  Gyy  Gz  be  the  moments  round  the  space-axes  of  the 
applied  forces,  and  Y  the  force  function,  we  have  then,  as  the 
three  equations  of  motion  of  the  body, 


—  (Ail  sin2  9  +  Cwz  cos  0)  =  Gz 

A  («A2sin2  9  +  92)  +  CV  =  2Y  +  constant 

C  —jj-  =  N  =  sin  9  (Gx  cos  \j/  +  Gy  sin  «A)  +  Gz  cos  9 


>■  (54) 


We  may  if  we  please  substitute  0  +  ^  cos  9  for  w3  in  (52), 
(53),  and  (54). 

Equations  similar  to  (54)  hold  good  for  a  free  body  if 
two  of  its  principal  moments  of  inertia  at  its  centre  of  inertia 
be  equal.  In  this  case  OZ  is  a  parallel  through  the  centre 
of  inertia  to  a  line  fixed  in  space. 

Examples. 

1 .  A  homogeneous  solid  of  revolution  terminating  in  a  cone  is  placed  with 
the  vertex  of  the  cone  on  a  perfectly  rough  horizontal  plane,  the  initial  condi- 
tions heing  given,  find  the  equations  of  motion. 

Here  the  vertex  of  the  cone  is  the  fixed  point  0 ;  and  if  a  vertical  line 
through  0  he  taken  as  the  space-axis  OZ,  since  gravity  is  the  only  force,  G- 
and  N  are  each  zero.  Then  hy  (54)  we  have  u>%  —  constant  =  w,  and  therefore 
the  first  two  of  equations  (54)  become 

Aty  sin2  6  +  On  cos  6  =  X,    A  (^2  sin2  0  +  02)  +  On2  =  E  -  2mgh  cos  0,     (a) 

where  h  is  the  distance  of  the  centre  of  inertia  from  0. 

If  we  take  a  point  P  on  00  the  axis  of  revolution  at  a  distance  I  from  0 
such  that  mhl  =  A,  this  point  P  is  the  centre  of  oscillation  of  the  body  for  an 
axis  perpendicular  to  ZOO.  Assuming  XI  =  Ona,  and  {E  -  On2)  I  =  Imghb,  we 
have,  then,  to  determine  the  motion  of  Pthe  equations 


mhP\p  sin20  =  On  (a  -  I  cos  6) 
J2(ipsin20  +  02)=  2g(b 


DS0)         V.  (b) 

I  cos  0)   ; 


Examples.  391 

2.  Give  a  geometrical  construction  for  the  velocity  of  P  in  any  position. 
Take  on  OZ  two  points  D  and  F  such  that  OB  =  a,  OF=b,  and  let  x  he  the 

angle  which  FD  makes  with  the  plane  of  the  horizon,  then  at  any  instant  the 
velocity  of  F  is  that  due  to  the  depth  of  F  helow  the  horizontal  plane  through 
F,  and  the  component  of  this  velocity  perpendicular  to  the  plane  ZOO  is 

On 

3.  Show  that  the  motion  of  the  axis  of  revolution  may  he  represented  by 
that  of  the  conjugate  line  (Art.  270)  in  a  body  not  acted  on  by  any  force. 

The  equations  of  motion  of  00  in  Ex.  1  are  of  the  same  form  as  those  of 
the  conjugate  line  in  Ex.  3,  Art.  270,  and  by  properly  determining  the  dis- 
posable constants  in  the  latter  they  may  be  made  identical  with  the  former. 

This  theorem  was  first  given  by  Jacobi,  but  the  mode  of  investigation  here 
adopted  is  due  to  Dr.  Routh. 

4.  Determine  the  limits  of  the  inclination  of  the  axis  of  revolution  to  the 
vertical. 

Eliminating  ^  from  equations  (b)  of  Ex.  1  we  obtain 


(Cn\2  (a-  lcos0\- 
\mh}     \    IsmQ     ) 


r-p  =  2?(b-uoSe)-(^-)   r-T^P)  .  (a) 


Now  when  0  attains  its  limiting  value,  0  =  0,  and  therefore  to  determine  the 
limiting  values  of  0  we  have  the  equation 

1A2g  {b  -  I  cos  0){1  -  cos2  0)  -  CPn2  (a  - 1  cos  0)2  =  F  (cos  0)  =  0.       (b) 

From  equation  (a)  it  is  plain  that  F  cannot  be  negative  for  any  value  of  0 
attained  in  the  actual  motion  of  the  body.  Hence,  if  i  be  the  initial  value  of 
0,  F  is  positive  or  zero  when  cos  0  =  cos  i.  Again,  it  is  easy  to  see  that  F  is 
negative  for  cos  0  =  —  1,  or  cos  0  =  1,  and  positive  for  cos  0  =  oo  .  We  con- 
clude that  the  equation  F  (cos  0)  =  0  has  three  real  roots,  two  between  - 1 
and  +  1,  and  one  between  +  1  and  oo  .  This  last  root  is  an  impossible  value  for 
cos  0.  In  general,  then,  the  angle  0  oscillates  between  two  limiting  values 
a\  and  ci2,  one  less  and  the  other  greater  than  the  initial  value  i.  That  this 
oscillation  should  be  possible,  it  is  necessary,  however,  that  F  should  vanish 
before  any  point  of  the  body  above  the  point  of  support  comes  into  contact  with 
the  horizontal  plane.  If  &  be  the  value  of  0  for  the  position  in  which  such 
contact  takes  place,  in  order  that  an  oscillation  should  be  possible,  i^cosjS)  must 
he  negative,  and  therefore 

C2 n2  (a  -  I  cos  £)2  >  1A2g  (b  -  I  cos  0)  sin2£.  (c) 

In  terms  of  the  original  constants,  iT  and  F,  this  condition  becomes 

{A  sin2 £  +  C cos2 0)  On2 -  2KCn  cosP>A(F-  2mgh  cos  ff)  sin2£ -  K2.     (d) 

5.  Show  that  the  character  of  the  oscillating  motion  depends  on  the  relative 
magnitudes  of  a  and  b. 


392  Kinetics  of  a  Rigid  Body. 

If  in  equation  (a)  Ex.  4,  we  make  I  cosO  -  a,  we  get  l202  =  2g(b-  a). 

If  a  be  less  than  b  this  gives  the  value  of  0  when  ty  =  0  ((b)  Ex.  1.)  In 
this  case  the  angular  motion  of  the  plane  ZOO  changes  its  direction  at  the 
point  corresponding  to  a  =  I  cos  0. 

Again,  if  a  >  b,  the  relation  a  =  I  cos  G  leads  to  an  imaginary  value  for  0, 
and  consequently  ^  cannot  vanish  during  the  motion.  Hence  in  this  case  the 
axis  00  rotates  constantly  in  one  direction  round  the  vertical  line  OZ. 

6.  Determine  in  any  particular  case  of  the  motion  whether  a  or  b  is  the 
greater. 

If  a  >  b,  then,  by  Ex.  1,:     —  > — . 

'    :  ,J     Cn        2mgh 

7.  If  the  axis  of  revolution  rotate  constantly  in  one  direction  round  the 
vertical,  and  if  ^0  be  the  value  of  \p  which  corresponds  to  either  the  greatest  or 
.  ,         -  ,        •         2mgh 

least  value  of  0,  prove  that  i|/0  <  — — . 

8.  Find  the  conditions  which  must  be  fulfilled  in  order  that  the  motion  of 
OC  should  be  steady. 

In  this  case,  if  it  can  occur,  the  inclination  of  00  to  the  vertical  and  the 
angular  velocity  of  the  plane  ZOO  are  constant.  If  we  eliminate  0  between  the 
two  equations  obtained  by  differentiation  from  (a),  Ex.  1,  we  get 

A0  =  Aip2  sin  0  cos  0  —  Cnty  sin  9  +  mgh  sin  0.  (a) 

Hence  if  0  =  0,  we  have 

sin0  (Afr  cos  0  -  C>4  +  mgh)  =0.  (b) 

If  sin  0  =  0,  we  have  0  =  0.  In  this  case  the  axis  is  vertical  throughout 
the  motion.     Again,  if 


we  obtain 


A  cos  0ty2  —  Cn^  +  mgh  =  0, 
On  ±  V(C2«2  -  4Amgh  cosfl) 


2 A  cos  0 

If  then  i  the  initial  value  of  0  fulfil  the  condition 

O2  n2  >  4Amgh  cos  i, 

and  if  likewise  initially  0  =  0,  and 

On  +  \J(C2 n2  —  4:Amgh  cos  i) 
*  =  ^ 2A7oTi '  {C) 

all  the  successive  differential  coefficients  of  0  and  ^  must  vanish  initially,  as 
readily  appears  from_  the  expression  for  0  and  the  first  of  equations  (a),  Ex.  1, 
and  therefore  0  and  ^  remain  constant,  and  the  motion  is  steady. 

9.  Prove  that  if  the  motion  be  not  steady  initially  it  cannot  become  so  sub- 
sequently. 

In  order  that  the  motion  should  become  steady  it  would  be  necessary  that 
0  and  0  should  vanish  simultaneously.  0  is  given  in  terms  of  0  by  equation  («), 
Ex.  4,  which  is  of  the  form  k02  sin20  =  i^cos  6),  where  h  is  constant.     If  we 


Examples.  393 

differentiate  this  equation,  divide  both  sides  by  0  sin  G,  and  then  make  0  and  0 
each  zero,  we  obtain  F'  (cos  0)  =  0.  Hence  if  0  and  0  vanish  together,  the  equa- 
tion i^(cos  0)  =  0  must  have  equal  roots.     Now  (Ex.  4), 

_F(cos  0)  =  2A-gl  (cos  0  -  cos  a\)  (cos  0  -  cos  a2)  (cos  0  -  A), 

where  A  is  always  greater  than  1.  Hence  if  the  equation  .F(cos  0)  =  0  have 
equal  roots,  we  must  have  ai  =  a2,  and  as  0  in  the  actual  motion  always  lies 
between  ai  and  a-,  the  double  root  must  be  cos  i,  where  i  is  the  initial  value 
of  0.  Consequently,  if  the  motion  be  not  steady  originally  it  can  never  become 
so. 

10.  A  peg-top  is  set  spinning  on  a  rough  plane,  determine  the  motion. 

In  this  case  the  only  initial  motion  is  a  rotation  round  OC,  and  therefore, 
if  i  be  the  initial  value  of  0,  we  have 

K=  Cn  cos  i,     E  -  Cn2  =  2mgh  cos  i. 

Hence  a  =  b  =  I  cos  i,  and  equations  (b)  Ex.  1,  become 

mhty  sin20  =  Cn  (cos  •  -  cos  0),    l{&  sin20  +  02)  =  2g  (cos  %  -  cos  9) .     {a) 

The  latter  equation  shows  that  cos  i  >  cos  0,  and  therefore  that  ty  has  the 
same  sign  as  ».  Hence  the  rotation  of  the  top  round  its  axis  is  in  the_  same 
direction  as  the  rotation  of  the  latter  round  the  vertical  through  the  point  of 
support. 

Again,  if  we  put  C2n-  =  IvmghA  =  ±vm- gh*  I,  equation  {b),  Ex.  4,  becomes 

(cosi  -  cos  0)  {1  -  2»/cosi  +  2v  cos  0  -  cos20}  =  0.  {b) 

Hence  0  =  i  or  i',  where  i'  is  determined  by  the  equation  sin2  i'  =  2v  (cos  i  -  ops  »'), 
and  0  oscillates  between  its  least  value  i  and  its  greatest  value  i',  provided  V  <  &, 

that  is  2v  >  ^-^ .     It  is  plain  that  the  latter  equation  is  what  (c)  Ex.  4 

cos  i  —  cos  j8 
becomes  in  the  present  case. 

1 1 .  Show  that  steady  motion  is  impossible  in  the  case  of  a  top,  except  the 
initial  position  of  its  axis  be  vertical. 

12.  Investigate  the  small  oscillations  of  the  axis  of  a  top  about  its  mean 
inclination  to  the  vertical. 

We  have  seen  that  if  v  or  n  fulfil  the  condition  given  at  the  end  of  Ex.  10, 
0  vanishes  when  0  =  i,  and  also  when  0  =  *'.  At  some  position  of  the  axis  of 
revolution  intermediate  between  these  two  0  =  0.  If  zs  and  a  be  the  values 
of  \p  and  0  corresponding  to  this  position,  we  have 

A^cos  a  —  CnZS  +  mgh  =  0  ) 

Azj  sin2a  +  Cn  cos  a  =  Cn  cos  i     ) 

The  motion  would  now  be  steady  if  0  were  zero.  .We  have  seen  that  this  is 
impossible ;  but  as  0  is  now  of  the  opposite  sign  to  0  the  axis  of  the  top  will 
oscillate  about  this  position,  provided  0  is  small. 


394  Kinetics  of  a  Rigid  Body. 

To  determine  these  oscillations,  let  \p  =  "&  +  <r,  0  =  a  +  e.  Then,  c  and  e 
being  small,  by  substituting  in  (a)  Ex.  8,  and  in  the  first  of  equations  (a)  Ex.  1, 
we  have 

d2e 
A  —  =  {Az32(cos,2a-sm2a)  +  (mgh  -  Cms)  cos  a}e+(2^3Jsina  cosa-Cto  sina)cr, 

Act  sin2o  +  sin  a  (2Azj  cos  a  -  Cn)  e  =  0. 

Substituting  for  ?h^  —  CnT3  its  value  given  by  (a),  we  get  from  the  first  of 

these 

d2e 
A  jy  +  ira26  sin2a  =  (2Azj  cos  a  —  Cn)  <r  sin  a ; 

but  from  the  second  we  have 

A<r  sin a  =  (Cn  —  2Azs  cos  a)  e, 

whence,  eliminating  <r,  we  obtain 

rf2€       ^42,S72  sin2a  +  (2Azj  cos  a  —  Cn)2 
dfi+ A> €=0' 

and  therefore  e  =j  sin  (/xt  +  7),  where  j  and  7  are  arbitrary  constants,  and  jx  is 
given  by  the  equation 

A2fi2  =  A2  zf  sin2o  i-  (2^4ot  cos  o  -  Crc)2. 

From  the  expression  for  82  given  by  (a)  Ex.  4,  it  is  easy  to  see  that  by 
properly  determining  n,  we  can  make  0  small  throughout  the  motion,  and  thus 
the  condition  requisite  for  a  small  oscillation  can  be  secured. 

13.  Find  the  vertical  pressure  on  the  plane  on  which  the  top  is  spinning. 
If  z  be  the  vertical  coordinate  of  the  centre  of  inertia  of  the  top,  and  P  the 
vertical  force  exerted  on  the  top  by  the  plane,  we  have  P  =  tug -\-  m'z ;  but 

..      ,   d   (dz\2  d       I  .        d9\2 

z  =  h  -r  (  —  I    —hn sin  6  —  )   , 

2  dz\dt  )       2     d  cos  0  \  dt  )  ' 

and  from  (a)  Ex.  4  and  Ex.  10,  we  have 

sin20  02  =  ^  (cos  i  -  cos  0)  { 1  -  2v  cos  i  +  2v  cos  0  -  cos20} . 

Hence        P=mg  J 1  +  -  (3  cos^  -  2  (cos  i  +  2v)  cos  0  +  4j/  cos  i  -  1)  j . 

14.  A  solid  of  revolution,  having  a  great  angular  velocity  round  its  axis,  and 
terminated  by  a  spherical  surface  of  small  radius,  is  placed,  with  its  axis  inclined 
to  the  vertical,  on  a  rough  horizontal  plane.  The  moment  of  inertia  round  the 
axis  of  revolution  being  not  less  than  that  round  an  axis  perpendicular  thereto, 
and  the  distance  of  the  centre  of  inertia  from  the  lower  end  being  considerable, 
show  that  after  some  time  the  axis  of  revolution  will  become  vertical.  (Jellett, 
Theory  of  Friction,  Chapter  VIII.) 


Examples. 


395 


Let  the  axis  of  z  through  the  centre  of  inertia  0  he  vertical,  and  letOC  be 
the  axis  of  revolution,  which  must  pass  through  S,  the  centre  of  the  terminating 
spherical  surface.     Accordingly  the  point  of  contact  T  lies  in  the  plane  ZOS. 

The  forces  acting  on  the  body  are  gravity,  and  the  resultant  of  the  normal 
reaction  and  friction  at  T.     The  friction  may  he  re-  z 

solved  into  two,  one  along  TZ',  the  other  at  right 
angles  to  the  plane  ZOC.  Calling  this  latter  com- 
ponent F,  and  putting 

TS  =a,     SO  =  b,     ZOC  =  0, 

the  moments  of  the  applied  forces  round  OZ  and  OC 
are  respectively 

Fb  sine,  and  -Fa  sin  0. 
Hence,  by  (54), 


—  (a  sin20tH  Co*  cos  0  J  =  Fb  sin  0 ; 


also, 


C =  -  Fa  sin 

at 


therefore 


-I 

at  \ 


A  sin-0^+Ca>3  cos  9 


)  ~~a      dt 


Hence,  putting    n  =  -, 

A  sin2 9\p  +  Ca>3 cos  9  +  nCwz  =  constant  =  CD.  (cos  0O  +  w), 

where  Q  and  90  are  the  initial  values  of  03  and  9,  since  ^  =  0  initially. 

As  the  force  F  constantly  diminishes  the  angular  velocity,  after  some  time 
a>3  must  become  equal  to 

n  +  cos  0o 

fi — . 

n+  1 


When  this  happens,  we  have  0  =  0.    For,  substituting  in  the  previous  equation 
the  value  just  obtained  for  a>3,  we  get 


(1 


(  „        .  n  +  cos  0o ) 

0)     2^cos2i0^  -  Cn  — - — j—  J  =0; 


but  as  «  is  greater  than  1,  ^  small  as  compared  with  n,  and  C  not  less  than  A, 
the  second  factor  of  the  above  expression  cannot  vanish,  and  therefore  we  must 
have  0  =  0.  .      , 

The  result  obtained  here  may  be  regarded  as  holding  good  in  the  case  01 
a  humming-top. 


396 


CHAPTER  XII. 

ENERGY  AND  THE  GENERAL  EQUATIONS  OF  DYNAMICS. 

Section  I. — Energy. 

280.  Energy. — Work  and  Energy  have  been  defined, 
Arts.  118  and  129,  and  the  equation  of  Energy  for  a  rigid 
body  has  been  obtained  by  two  different  methods  (Arts.  132 
and  200).  In  the  present  section  we  propose  to  consider 
the  subject  in  a  somewhat  more  general  manner,  and  to  show 
that  on  the  equation  of  Energy  may  be  based  the  whole 
theory  of  the  action  of  forces  on  a  connected  system. 

281.  Equation  of  Energy. — If  a  system  of  material 
points  be  acted  on  by  any  forces,  we  may  suppose  the  con- 
straints and  connexions  of  the  system  replaced  by  correspon- 
ding forces,  and  thus  regard  each  point  as  entirely  free. 
Assuming  then  the  principles  which  govern  the  resolution 
and  composition  of  forces  acting  at  a  point,  and  the  relations 
between  force  and  acceleration,  we  have 

d2xx  d2yx 

mW  =  x»    m'UF 

d2x2  d2y2 

"h^iF  =  x'-'    "hHF 

where  Xi,  YXi  Zx,  &c,  include  the  components  of  the  forces 
which  replace  the  constraints,  if  any,  acting  at  the  points 

Multiplying  the  first  equation  by  dxly  the  second  by  dyly 
&c,  and  adding,  we  have 

(d2x    ,       d2i/    ,        d2z  _  \      CT/^7        t,  ,        m\ 
2  m(  — -  dx  +  -^  dij  +  —  dz  J  =  2{Xdx  +  Ydy  +  Zdz) ; 


F„ 

a  Z]_ 

Y,, 

"hle=z"" 

Conservation  of  Energy.  397 

or,  putting  T  =  J  2m#2, 

— r .  dt  =  2  (X<te  +  Ftfy  +  Zflfe).  (1) 

In  virtue  of  this  equation,  T  is  called  the  kinetic  energy  of 
the  system  (Art.  129). 

282.  Conservation  of  Energy. — If  the  mutual  forces 
of  a  material  system  are  independent  of  the  velocities,  the  system 
must  be  conservative  (Art.  124). 

To  prove  this  we  have  to  show  that,  in  going  from  any 
configuration  A  to  the  configuration  B,  the  work  done  by 
the  forces  of  the  system  is  independent  of  the  mode  of  trans- 
formation, and  depends  only  on  the  initial  and  final  con- 
figurations. 

Suppose  that  in  one  mode  M  of  going  from  A  to  B  more 
work  is  done  than  in  another  mode  N.  Let  us  imagine  two 
systems  precisely  similar,  and  let  the  first,  going  from  A  to  B 
by  the  mode  M,  be  made  to  bring  the  other  from  B  to  A  by 
the  mode  JY.  This  will  be  possible,  because  the  work  consumed 
in  going  from  B  to  A  through  N  is  equal  to  the  work  per- 
formed in  going  from  A  to  B  through  JY,  since  the  forces  in 
any  particular  position  are  by  hypothesis  independent  of  the 
directions  in  which  the  points  of  the  system  are  moving, 
and  therefore  each  element  Xdx  of  the  total  work  retains  the 
same  magnitude,  but  changes  its  sign,  when  the  tranforma- 
tion  is  reversed.  Hence  the  transformation  from  A  to  B 
through  M  will  bring  the  second  system  from  Bio  A  through 
i\T,  and  leave  an  overplus  of  work.  Let  us  now  suppose  the 
second  system  to  go  from  A  to  B  through  M,  bringing  the 
first  from  B  to  A  through  N.  There  will  again  be  an  over- 
plus of  work.  This  process  may  be  continually  repeated, 
and  thus  an  inexhaustible  supply  of  work  can  be  obtained 
from  permanent  natural  causes  without  any  consumption  of 
materials.  The  whole  of  experience  teaches  us  that  this  is 
impossible.  Hence  the  work  done  in  going  from  A  to  B  is 
independent  of  the  mode  of  transformation. 

If  a  system  be  acted  on  by  no  external  forces,  the  work 
done  by  the  forces  of  the  system  is  equal  to  the  change  of 
kinetic  energy  ;  whence  it  appears  that  the  kinetic  energy  T 


398  Energy. 

in  any  particular  configuration  depends  only  on  the  values  of 
the  variables  by  which  the  configuration  is  indicated,  and  on 
the  initial  state  :  in  other  words,  we  have  the  equation 

T  -  T0  =  (p  (a*,  yx,  «i,  0S2,  y2,  s2,  &c).  (2) 

It  is  essential  to  the  validity  of  this  demonstration  that 
the  work  consumed  by  the  forces  of  the  system,  in  any  trans- 
formation, should  be  equal  to  the  work  performed  by  them  in 
the  same  transformation  reversed.  If  the  force  acting  on  any 
point  changes  sign  with  the  direction  of  the  motion  of  that 
point,  the  condition  of  reversibility  is  not  fulfilled.  In  the 
case  of  friction,  for  instance,  so  far  as  it  is  considered  in 
Mechanics,  the  forces  change  sign  with  the  motion,  and 
consume  work  both  in  the  direct  and  reverse  transformation. 
The  same  is  true  of  the  resistance  of  a  medium.  Again,  if 
the  forces  are  not  equal  in  magnitude  when  the  points  occupy 
the  same  relative  positions,  as  in  the  case  of  the  collision  of 
imperfectly  elastic  bodies,  work  is  apparently  consumed  with- 
out any  corresponding  increase  of  potential  energy.  The  ex- 
periments of  Joule  and  others  have  established  that  in  such 
cases  the  energy  which  seems  to  be  lost  is  really  preserved  in 
the  form  of  heat,  which  may  be  regarded  as  kinetic  energy 
resulting  from  molecular  motions  not  directly  sensible.  In 
applying  the  equation  of  energy  we  must,  however,  remember 
that  in  cases  such  as  those  mentioned,  the  conservation  of 
energy,  so  far  as  sensible  motion  alone  is  concerned,  does  not 
hold  good.  On  the  other  hand,  if  we  take  into  account  every 
form  of  energy,  the  conservation  of  energy  may  be  considered 
as  an  absolutely  universal  fact  of  nature. 

Equation  (2)  may  be  written 

T-T0=Y-Yo,  (3) 

where  Y  is  the  force  function  given  by  the  equation 

Y  =  ^{Xdx  %  Ycly  +  Zdz). 

Equation  (3)  is  the  same  as  (12),  Art.  200,  where  it  was 
arrived  at  in  a  different  manner. 
If  we  put  Y  =  -  V,  we  have 

T+V=T0+Vo.  (4) 

We  can  now  give  an  exact  definition  of  potential  energy. 


Of  the  Ultimate  Permanent  Forces  of  Nature.         399 

The  Potential  energy  of  a  conservative  system  in  any  par- 
ticular configuration  is  the  amount  of  work  required  to  bring 
it  to  that  configuration  against  the  mutual  forces  of  the  system 
in  its  passage  from  any  chosen  configuration. 

The  principle  of  the  conservation  of  energy  may  then  be 
stated  thus :  — 

In  any  conservative  system  unacted  on  by  external  forces  the 
sum  of  the  kinetic  and  potential  energies  is  constant. 

283.  Of  the  Ultimate  Permanent  Forces  of  Mature. 

— In  his  Paper  on  the  Conservation  of  Force  {TIeber  die 
Erhattung  der  Kraft,  1847),  Helmholtz  observes  that  we 
must  regard  the  forces  of  nature  as  caused  by  the  action  of 
portions  of  matter  on  each  other,  and  from  a  mathematical 
point  of  view  must  consider  matter  as  composed  of  an  infi- 
nite number  of  material  points.  The  ultimate  permanent 
forces  of  nature  must  result,  therefore,  from  the  action  of 
these  material  points  on  each  other. 

If  the  conservation  of  energy  hold  good  for  these  forces, 
the  mutual  forces  between  two  material  points  must  be  in  the 
line  joining  them,  and  be  a  function  of  the  distance  between 
them. 

This  proposition  is  proved  by  Helmholtz  as  follows : — 
In  this  case  the  kinetic  energy  of  the  system  composed  of 
the  two  points  is  given  by  the  equation 

T=  j  [Xda>  +  Ydy  +  Zdz  +  X'dx  +  Ydy  +  71  dz)  +  c. 

Since  the  conservation  of  energy  holds  good,  T  is  a 
function  of  the  relative  position  of  the  two  points.  Again,  as 
they  are  points,  all  directions  must  be  supposed  indifferent  as 
regards  either  of  them  considered  alone.  Hence  their  relative 
position  must  depend  solely  on  the  distance  between  them, 
and  T  is  therefore  a  function  of  this  distance  r,  or  T  =  $  (r). 

Equating  the  two  expressions  for  T,  and  differentiating, 
we  have 

X-#'W£     F  =*'(,■)  g,    Z=¥(r)%, 
X-fW*      Y'-¥ir)%,    W(r)*; 


400  Energy. 

or     X-#'(r)fZ^      r_^(r)LZ*      Z=<j>'(r)~, 


X'=0'(r)^,     T-f\r)t-X    Z'=4>'(r) 


r 


Hence  the  point  xyz  is  acted  on  by  a  force  <j>'(r)  in  the 
direction  of  the  line  joining  xyz  to  ocyz  ;  and  the  latter  point 
is  acted  on  by  an  equal  force  in  the  opposite  direction. 

Conversely  it  is  easy  to  see  that,  if  two  material  points 
acted  on  each  other  with  a  force  depending  as  regards  mag- 
nitude on  their  mutual  distance,  but  not  in  the  direction  of  the 
line  join  in  cj  them,  they  would  be  capable  of  producing  in  each 
other  an  ever-increasing  velocity,  and  of  thus  generating  an 
unlimited  amount  of  energy. 

In  order  to  bring  about  this  result  we  have  only  to  suppose 
the  points  connected  by  a  rigid  rod.  The  whole  system  would 
then  be  acted  on  by  a  constant  couple. 

284.  Forces  which  appear  in  the  Equation  of 
Energy. — For  any  system  entirely  free  wre  have  obtained 
the  equation  dT  =  S(Xda?  +  Tdy  +  Zdz),  and  have  seen  that 
this  equation  holds  good  for  a  system  restricted  in  any  way, 
provided  the  constraints  are  replaced  by  equivalent  forces. 

If  the  constraints  of  the  system  consist  of  smooth  curves 
or  surfaces  along  which  the  points  are  restricted  to  move  ;  of 
rigid  bars  or  inextensible  strings  connecting  the  different 
points  with  each  other  or  with  any  external  fixed  points  ;  or 
in  general  of  any  connexions  such  that  the  distance  between 
each  pair  of  points  immediately  acting  on  each  other  is  in- 
variable, the  whole  work  done  by  all  these  constraints  and 
connexions  is  zero,  and  may  therefore  be  omitted  from  the 
right-hand  side  of  the  equation  (Arts.  124,  127). 

If  the  potential  energy  (Arts.  129,  282)  of  any  portion  of 
the  system  be  a  function  of  a  single  variable  quantity  u9  the 
work  done  by  this  part  of  the  system  in  any  displacement 
must  be  of  the  form  ASw;  for  V=  <f>  (u),  and  therefore 
dV  =  <j>'(u)du. 

If  between  any  points  of  the  system  there  be  a  connexion 
which  is  capable  of  being  expressed  by  means  of  an  equation 


Forces  which  appear  in  Equation  of  Energy.  401 

"between  their  coordinates,  such  connexion  can  be  effected  by 
means  of  constraints  of  an  invariable  character ;  such  as  smooth 
fixed  surfaces  or  curves,  or  rigid  bars  or  inextensible  strings. 

Hence  we  may  conclude  that,  in  any  motion  of  the  system, 
the  work  done  by  the  forces  replacing  any  connexion  between  the 
points  of  the  system  which  is  capable  of  being  expressed  by  equa- 
tion* between  their  coordinates,  is  zero. 

A  formal  proof  of  this  important  proposition  may  be  given 
as  follows : — 

If  TJ  =  0  be  an  equation  between  the  coordinates  of  any 
points  in  a  moving  system,  the  forces  which  the  corresponding 
constraint  introduces  into  the  system  must  be  functions  of 
the  coordinates  and  of  the  other  forces.  Hence,  if  the  latter 
be  conservative,  so  are  the  forces  caused  by  the  constraint, 
which  for  brevity  we  shall  refer  to  as  the  constraint  TJ. 

Again,  if  at  any  time  the  condition  TJ-  0  be  actually  ful- 
filled, the  imposition  or  removal  of  the  material  bonds  by 
which  the  corresponding  constraint  is  effected  cannot  require 
any  expenditure  of  energy  ;  since  this  imposition  or  removal 
does  not  change  the  position  of  any  point  of  the  system. 

Let  there  be  now  a  system  Si9  which  without  27 is  conser- 
vative, and  let  A  and  B  be  two  configurations  in  which  the 
condition  U  =  Q  is  fulfilled  ;  then,  as  we  have  seen,  the  forces 
replacing  £7  are  conservative,  and  if  they  consume  work  in  the 
motion  from  B  to  A,  they  produce  work  in  that  from  A  to  B. 
Let  the  external  work  W  bring  Si  from  A  to  B  subject  to 
the  constraint  TJ.  Let  Q  be  the  amount  of  potential  energy 
thereby  produced  in  the  system,  and  E  the  work  clone  by  the 
forces  replacing  U;  then,  the  whole  amount  of  work  produced 
is  W  -  Q  +  E.  Now  let  this  be  used  in  bringing  S2  (precisely 
similar  to  Sy)  from  B  to  A  without  the  constraint  TJ,  whereby 
Q  is  produced,  and  in  doing  such  an  amount  of  other  work 
that  Si  may  come  to  rest  in  the  position  B  and  S2  in  the 
position  A.  We  may  then  without  any  expenditure  of 
work  impose  the  constraint  TJ  on  S%  and  remove  it  from  Si. 
Things  are  now  in  precisely  the  same  state  as  at  starting,  and 
in  the  whole  process,  by  an  expenditure  of  work  W,  we  have 
produced  work  whose  amount  is  W  +  E.  Hence  in  any 
motion  of  the  system  the  work  E  done  by  the  forces  replac- 
ing the  condition  TJ  =  0  must  be  zero. 

2  1) 


402  Energy. 

As  the  amount  of  work  done  by  these  forces  in  an  un- 
reversed motion  cannot  be  influenced  by  the  character  of  the 
other  forces,  but  only  by  their  amounts  and  directions,  the 
work  done  by  the  forces  replacing  U  =  0  must  under  any 
circumstances  be  zero. 

285.  Equation  of  Energy  in  General. — If  we  have  a 
system  acted  on  by  any  forces  external  or  internal,  and  sub- 
ject to  any  constraints  or  mutual  connexions,  the  equation  of 
energy  assumes  the  form 

T-T0+  V-  V0=  W.  (5) 

T0  and  V0  are  the  kinetic  and  potential  energies  in  the 
initial  position,  T  and  V  those  in  the  position  under  con- 
sideration, and  W  the  work  done  in  going  from  the  initial 
to  the  actual  position  by  the  external  forces  and  by  those 
internal  forces  which  are  not  conservative  or  reversible  in  their 
character. 

As  regards  constraints  and  connexions,  they  may  be  di- 
vided into  three  classes.  1.  Those  producing  forces  whose 
work  during  any  motion  of  the  system  is  zero.  Such  con- 
nexions we  have  already  considered  ;  they  have  no  effect  on 
the  equation  of  energy.  2.  Those  which  are  capable  of  alter- 
ation under  the  action  of  external  forces,  and  such  that  their 
alteration  produces  or  consumes  a  corresponding  amount  of 
potential  energy.  The  work  done  by  the  forces  replacing 
these  constraints  and  connexions  is  included  in  the  expression 
Vq  -  V.  3.  Resistances  or  connexions  which  introduce  forces 
of  a  non-conservative  character.  Such  are  the  friction  of 
rough  surfaces,  the  resistance  of  a  medium,  the  forces  deve- 
loped by  the  alteration  of  an  extensible  body  which  does  less 
work  in  its  recoil  than  the  amount  required  to  stretch  it,  &c. 
All  such  forces  must  appear  as  forces  in  the  equation  of 
energy,  and  the  work  done  by  them  is  included  in  W. 

286.  Virtual  Velocities. — The  principle  of  energy  may, 
as  we  have  seen,  be  expressed  for  dynamical  purposes  in  the 
following  form  : — 

The  work  done  on  a  system  in  any  interval  of  time  by 
external  applied  forces,  diminished  by  the  work  consumed  in 
the  same  time  by  the  non- conservative  forces  of  the  system, 
is  equal  to  the  sum  of  the  increments  of  the  kinetic  and 
potential  energies. 


Virtual  Velocities.  403 

We  have  seen  likewise  that  this  principle  holds  good  for 
a  system  subject  to  any  invariable  constraints  or  connexions 
internal  or  external  as  well  as  for  a  free  system. 

We  are  now  able  to  obtain  the  conditions  which  must  be 
fulfilled  in  order  that  any  system  should  be  in  equilibrium ; 
they  can  be  expressed  in  a  single  statement,  viz  : — 

In  order  that  any  system  should  be  in  equilibrium,  the  ivork 
done  by  the  applied  forces  in  any  possible  infinitely  small  displace- 
ment, diminished  by  the  increase  of  the  potential  energy  of  the 
system,  must  be  cither  negative  or  zero ;  and,  if  this  be  true 
for  every  possible  infinitely  small  displacement,  the  system  is  in 
equilibrium. 

The  truth  of  this  statement  readily  appears  from  the 
equation  of  energy. 

A  position  of  equilibrium  is  one  in  which  if  the  system  be 
placed  at  rest  it  will  remain  at  rest.  Now  the  system  will 
not  remain  at  rest  if  there  be  any  possible  mode  of  displace- 
ment, in  which  the  united  action  of  the  internal  and  external 
forces  can  produce  a  velocity  in  any  of  the  points  of  the  system. 
On  the  other  hand,  if  the  system  move  from  rest  in  any 
manner,  it  will  acquire  a  positive  kinetic  energy.  _  Hence,  if 
there  be  no  possible  way  in  which  it  can  do  this,  its  position 
must  be  one  of  equilibrium. 

In  applying  the  principle  of  equilibrium  we  must  regard 
the  non-conservative  forces  of  the  system  (if  any)  as  applied 
forces,  and  introduce  them  with  their  proper  signs.  In  the 
case  of  actual  motion,  forces  of  this  kind  always  consume 
work,  but  in  the  case  of  virtual  displacements  this  is  not  ne- 
cessarily the  case ;  e.  g.  suppose  a  heavy  particle  is  placed 
on  a  rough  inclined  plane,  and  it  is  required  to  determine  the 
condition  of  equilibrium.  In  this  case  we  must  consider  the 
force  of  friction  as  acting  upwards  along  the  plane.  If  now 
we  imagine  a  virtual  displacement  down  the  plane,  friction 
will  consume  work  ;  but  if  we  imagine  a  displacement  up  the 
plane,  friction  will  produce  work.  In  the  case  of  actual 
motion,  whether  slipping  take  place  up  or  down  the  plane, 
friction  will  consume  work. 

Again  it  is  to  be  observed,  that  if  every  possible  set  of 
displacements  be  also  possible  when  reversed,  the  condition  of 

2D  2 


404  Energy. 

equilibrium  becomes  simply  that  the  total  work  done  by  all  the 
forces  internal  and  external  be  zero. 

In  fact,  if  SPSp  be  negative  and  P  remaining  unaltered 
the  sign  of  each  §p  be  changed,  SPSp  becomes  positive  ;  but 
this  is  inconsistent  with  the  principle  of  equilibrium  as  stated 
above  ;  hence  ^P^p  must  be  zero. 

If  we  combine  the  principle  of  Virtual  Velocities  with 
D'Alembert's  principle,  we  obtain  the  equation  which  embraces 
the  whole  theory  of  Kinetics, 

From  this  equation  that  of  energy  was  deduced  in 
Chapter  IX.  In  the  present  chapter  we  have  reversed  this 
mode  of  procedure. 

287.  Equivalent  Sets  of  Forces. — Two  sets  of  forces 
acting  on  any  material  system  are  said  to  be  equivalent  when 
the  motions  produced  by  one  set  are  identical  with  those 
produced  by  the  other. 

If  each  of  two  sets  of  forces  be  capable  of  equilibrating  the 
same  third  set,  the  two  are  equivalent. 

For  let  P  be  a  force  of  the  first  set,  Q  one  of  the  second,  and 
R  one  of  the  set  which  each  of  the  first  two  can  equilibrate. 
Suppose  the  P  set  only  to  act.  Introduce  at  the  point  where 
R  would  act  two  forces  R  and  -  R.  This  being  done  for  each 
point  of  the  system,  the  motion  remains  undisturbed.  The 
system  is  now  acted  on  by  the  three  sets  of  forces  P,  P,  and 

-  R ;  and,  since  the  sets  P  and  R  are  in  equilibrium,  the  sets 
P  and  -  R  are  equivalent.     In  like  manner  the  sets  Q  and 

-  R  are  equivalent.     Hence  the  sets  P  and  Q  are  equivalent. 

In  moving  a  system  from  one  given  position  to  another,  equi- 
valent sets  of  forces  produce  the  same  amount  of  work. 

The  motion  being  the  same  whichever  set  of  forces  is 
in  action,  the  intermediate  positions  of  the  system  are  at  each 
instant  the  same ;  consequently,  since  the  two  sets  of  forces 
are  each  capable  of  equilibrating  the  same  set,  we  have  %P$p 
=  ^QBq  at  each  instant.  Hence  the  whole  amount  of  work 
produced  in  one  case  is  equal  to  that  produced  in  the  other. 

It  can  be  shown  in  like  manner  that  the  work  required  to 
move  a  system  from  one  given  position  to  another,  against  the 


Examples.  405 

action  of  any  set  of  forces,  is  equal  to  that  required  to  move  it 
against  the  action  of  an  equivalent  set. 

288.  "Wrenches. — A  wrench  in  Kinetics  corresponds  to 
a  twist  in  Kinematics. 

If  a  rigid  body  be  acted  on  by  any  forces,  these  forces  can 
be  reduced  to  a  single  force  along  with  a  couple  whose  plane 
is  perpendicular  to  the  direction  of  the  force. 

Such  a  system  is  called  a  wrench  about  a  screw,  the  axis  of 
the  screw  being  the  line  of  direction  of  the  force,  and  the  pitch 
of  the  screw  the  line  which  is  the  quotient  obtained  by  dividing 
the  moment  of  the  couple  by  the  force.  The  magnitude  of 
the  force  is  called  the  intensity  of  the  wrench. 

The  wrench  to  which  a  set  of  forces  acting  on  a  rigid 
body  is  equivalent  has  been  termed  the  canonical  form  of  the 
set  of  forces. 

The  canonical  form  of  a  set  of  forces  is  in  general  unique ; 
for,  as  may  be  easily  seen,  if  two  wrenches  be  equivalent, 
they  must  either  be  identical  or  else  consist  of  equal  couples 
in  parallel  planes. 

Examples. 

1 .  A  particle  of  mass  m  moves  with  a  simple  harmonic  motion ;  determine 
its  mean  energy. 

If  t  and  a  be  the  periodic  time  and  amplitude  of  the  motion  (Arts.  87,  88), 
and  T  the  mean  energy, 


1  rrmv2   ,  7r2   . 

T=~\    — -  dt  =  m  —  a2. 

2  T" 


r 

Jo 


2.  If  the  motion  of  the  particle  m  he  the  resultant  of  any  number  of  simple 
harmonic  motions  having  different  periods  and  amplitudes,  find  the  mean  value 
of  the  energy. 

If  0  be  an  interval  of  time  which  is  very  great  compared  with  the  longest 
periodic  time, 


-if* 


— -dt  =  W7T-2— . 

Z  T" 


3.  Determine  the  mean  energy  of  a  system  of  vibrating  particles. 

The  rectangular  components  of  the  displacement  of  any  particle  are  periodic 

functions  of  the  time,  and  can  therefore  be  expanded  in  a  series  of  terms  of  the 

form 

•    /2tt  \ 

asm    —  t  +  a  J . 

(i-  _l  yi  jl.  (<2 
Hence,  T=  Tr-Xm — -. 


406  Energy. 

4.  A  rigid  body  is  acted  on  by  a  couple  whose  moment  is  Pp ;  determine 
the  work  done  by  the  couple  in  any  small  motion  of  the  body. 

If  d9  be  the  angular  displacement  of  the  body  round  an  axis  perpendicular 
to  the  plane  of  the  couple,  the  work  done  by  the  couple   is  Ppdd,  see  Art.  128. 

5.  Express  the  kinetic  energy  of  a  body  having  a  fixed  point  in  terms  of 
the  angles  0,  cj>,  $  (Art.  258),  the  body-axes  being  the  principal  axes  at  the 
fixed  point. 

As  an,  «2,  co3  are  given  in  terms  of  0,  (f>,  »//,  Ex.  5,  Art.  260,  we  have, 
Art.  263, 

2T=  (^sin2<|)  +  -Bcos2^)02+{(^cos2(/)  +  ^sin2(/))sin30+Crcos20}  fr 

+  C <p2  +  2  (B -  A)  £ty  sine  sin  cp  cos  <f>+  2 C$^  cos0. 

6.  If  T  be  the  kinetic  energy  of  a  body  having  a  fixed  point,  and  <ax,  wy,  w3 
its  angular  velocities  round   three   rectangular   axes   fixed  in   space  passing 

dT  dT     dT 

through  the  point,  prove  that  - —  - — ,  - —  are  the  moments  of  momentum  of 

do)X  dwv  dooz 
the  body  round  the  axes. 

Let  x,  y,  z  be  the  components  of  the  velocity  of  any  point  of  the  body,  then 

dT  I     dx  dy  dz  \ 

2T=2m{xi  +  y*  +  z* ),    —  =  2»i  [x—  +  y  ■/-  +  s  —    ; 

hence  (Art.  255), 

—  =  2m(yz-zy). 
duox 

7.  Determine  the  amount  of  energy  W  which  must  be  expended  on  a  rigid 
body  in  order  to  effect  a  given  twist  in  opposition  to  a  given  wrench. 

Let  0  be  the  amplitude  of  the  twist  round  the  screw  a  whose  pitch  is  p, 
Q  the  intensity  of  the  wrench  round  the  screw  13  whose  pitch  is  q,  and  ii  the 
angle  between  a  and  /3. 

Take  a  as  axis  of  x,  and  the  shortest  distance  from  a  to  $  as  axis  of  z, 
denoting  its  length  by  c.  Eeplace  the  wrench  at  each  instant  by  the  forces 
X,  Y,  Z,  passing  through  a  point  coinciding  with  the  origin,  and  the  couples 
i,  M,  N.     Then  the  system  X,  Y,  Z,  L,  M,  N  being  equivalent  to  the  wrench, 

X=Qcosn,      Z=Qsinn,     Z=0, 

L  =  Qq  cos  a  -  Qc  sin  n,     M=  Qq  sin  D,  +  Qc  cos  n,     N=  0. 
Hence  (Ex.  4,  and  Art.  123), 
dW=  Qp  cos  ad9  +  [Qq  cos  n  -  Qc  sin  fl)  dd  =  Q  { (p  +  q)  cos  a  -  c  sin  n}  d0, 

and  therefore 

W=  Q  {{p  +  q)  cosn-  c  sinn}0. 


Examples.  407 

The  expression  (p  +  q)  cos  fl  -  e  sin  n  is  called  (Ball,  Theory  of  Scrcivs,  §  13) 
the  virtual  coefficient  of  the  pair  of  screws  a  and  /3. 

If  c  he  regarded  as  always  positive,  ft  is  the  angle  through  which  the  axis 
of  the  screw  a  must  he  turned  round  the  axis  of  z  in  order  to  he  codirectional 
with  the  axis  of  £,  the  positive  direction  of  z  heing  always  from  a  towards  j8. 

8.  A  smooth  rod  having  one  extremity  fixed  moves  on  a  smooth  horizontal 
table,  and  drives  a  particle  of  mass  equal  to  its  own,  which  starts  from  rest  from 
a  point  indefinitely  near  the  fixed  extremity  of  the  rod.  Find  the  inclination  of 
the  rod  to  the  direction  of  motion  of  the  particle  when  the  latter  has  reached  any 
definite  point  of  the  rod. 

As  the  moment  of  momentum  and  the  vis  viva  of  the  system  are  constant  {see 

Art.  201),  we  have,  if  i//  be  the  angle  required,  tan  ^  =  -— — — ,  where  k  is  the 

radius  of  gyration  of  the  rod  and  r  the  distance  of  the  particle  from  the  fixed 
extremity. 

9.  A  triangular  prism  rests  with  one  rectangular  face  on  a  smooth  horizontal 
plane.  A  rough  cylinder,  having  its  axis  parallel  to  the  edge  of  the  prism,  rolls 
down  one  of  its  faces  starting  from  rest,  the  centres  of  inertia  of  the  prism  and 
cylinder  being  in  the  same  vertical  plane  ;  determine  the  angular  velocity  of  the 
cylinder  when  it  reaches  the  horizontal  plane,  and  the  distance  through  which 
the  prism  has  moved. 

Let  the  axis  of  #  be  the  intersection  of  the  horizontal  plane  with  the  vertical 
plane  containing  the  centres  of  inertia,  the  axis  of  z  being  vertical.  Let  x  be 
the  coordinate  of  the  centre  of  inertia  of  the  prism,  m'  its  mass ;  x,  z  the  co- 
ordinates of  the  centre  of  the  cylinder,  a  its  radius,  m  its  mass,  k  its  radius  of 
gyration,  (p  the  angle  through  which  it  has  turned,  and  s  the  distance  on  the 
prism  perpendicular  to  its  edge  through  which  the  line  of  contact  of  the  cylinder 
has  moved,  at  any  time ;  then,  i  being  the  angle  which  the  face  of  the  prism 
makes  with  the  horizontal  plane, 

x  —  x  =  Xo  —  xo'  +  s  cos  i,     z  —  Co  —  s  sin  i,     s  =  a<p  ; 

and  from  the  conservation  of  the  motion  of  the  centre  of  inertia  and  the  equation 
of  vis  viva,  we  have 

mx  +  m'x'  =  mxa  -f  m'x0',     m'x2  +  m {x2  +  c2  -f  k2(f>2)  =  2gm  (z0  -  z). 

Hence,  if  h  be  the  initial  height  of  the  centre  of  the  cylinder,  and  a>  its 
angular  velocity  when  it  reaches  the  horizontal  plane, 

or  = ; — ,     x  -  Xo  =  —  ■ ;  (h  —  a)  cot  x. 

{m  +  m  sin2i)  a4  +  (m  +  m)  k"1  m  +  m 

10.  Show  that  the  velocity  v  with  which  a  fluid,  under  a  uniform  pressure 
p,  escapes  from  a  small  orifice  is  given  by  the  equation  v2  =  2gh,  where  h  is  the 
height  of  a  column  of  the  fluid  which  would  produce  the  pressure  p. 

Suppose  a  small  mass  m  of  fluid  forced  through  an  orifice,  whose  section  is  <r, 
into  a  large  volume  of  fluid  under  the  pressure  p.  If  x  be  the  distance  through 
which  the  surface  <r  of  the  fluid  is  pushed  in  by  this  operation,  the  work  ex- 
pended is  pxcr. 


408  Energy. 

Hence  the  potential  energy  lost  when  m  escapes  is  pxa,  and  this  must  be  the 
kinetic  energy  acquired  by  m,  therefore  —  =px<r.  Now,  if  p  be  the  density  of 
the  fluid,  p  =  ffph,  and  m  =  px<r.     Substituting,  we  have 

v-  =  2gh. 

1 1 .  Determine  the  total  energy,  kinetic  and  potential,  of  a  planet  and  satel- 
lite moving  as  in  Ex.  3,  Art.  213. 

If  T  be  the  total  energy 

where  K  is  an  undetermined  constant,  and  C,  &c,  have  the  same  significations 
as  in  the  example  referred  to. 

12.  A  planet  and  a  satellite  move  as  in  the  last  example.  If  with  a  given 
moment  of  momentum  it  is  possible  to  set  them  moving  as  a  rigid  body,  it  is 
possible  to  do  so  in  two  ways — for  one  of  which  the  energy  is  a  maximum,  and 
for  the  other  a  minimum. 

If  in  the  equation  of  Ex.  3,  Art.  213,  we  substitute  x  forr*,  y  for  n,  and  put 
h  for  the  moment  of  momentum,  we  have 

h  =  Cy  +  /j}Mm(M+  m)-±x. 
Again  (Ex.  11), 

2Y-K=  Cif-fxMm-- 
xl 

By  a  proper  selection  of  the  units  of  mass,  length  and  time,  we  can  make  C, 
/x*Mm(M+  m)-i,  and  /xMm  each  equal  to  unity.     We  obtain  thus  for  the  unit  of 

mass ,  for  that  of  length  <  — — \  ,  andfor  that  of  time!     „  „„    ,  >  , 

M+m  (      Mm      )  [  fi2M3m3  ) 

We  have  then 

h  =  y+x,     2Y-K=y2-—- 

If  the  whole  system  move  as  a  rigid  body,  the  angular  velocity  <a  of  the 
satellite  round  the  centre  of  inertia  of  the  system  must  be  equal  to  n ;  but 
r'A  =  ix  {M  +  m)co'2,  and  in  the  special  units  adopted  n*(M  +  m)~~  =  1  ;  hence 

xz  =  -.     Again,  if  Fbe  maximum  or  minimum,  we  have 

y 

—  \  (k  -  x)~  -  —  |  =  0,     or    x-h  +  —  =  0, 
dx  {  x~)  x3 

which  is  the  same  equation  for  determining  x  as  before.     Hence,  if  the  whole 
system  move  as  a  rigid  body,  the  total  energy  Zis  a  maximum  or  a  minimum. 

Again,  let  f(x)  =  z4  -  hxA  +  1  =  xs  (x  -  h)  +  1 .  If  x  be  negative  f[x)  is  positive, 
and  therefore  the  biquadratic  f(x)  =  0  has  no  negative  root,  and  cannot  therefore 
have  more  than  two  real  roots,  since  the  coefficient  of  x1  is  zero.  If  x>  h,  f(x) 
is  positive,  and  therefore  the  biquadratic  has  no  positive  root  greater  than  /* ; 


Examples.  409 

but  if  x  be  positive  and  less  than  h,  f(x)  may  be  negative,  and  therefore  the 
biquadratic  may  have  two  positive  roots  between  0  and  h.  Asf'(x)  =  x-(ix  —  37*), 
if  the  biquadratic  have  two  real  roots,  one  is  greater  than  f  A  and  the  other  less. 

The  greater  root  makes /'(#)>  and  therefore  —  positive,  and  Fa  minimum; 

d2Y 
the  lesser  root  makes  — -  negative,  and  F  a  maximum. 

13.  Apply  the  preceding  examples  to  determine  the  secular  effects  of  tidal 
friction  on  the  Earth-moon  system,  the  moon  being  supposed  to  move  in  the 
plane  of  the  equator. 

If  the  Earth's  radius  be  denoted  by  0,  Cis  approximately  f Ma\  and  M  =  XZm. 

Hence  the  unit  of  mass  is  f£,  the  unit  of  length  5-26*,  and  the  unit  of  time 

2  hours  41  minutes.  Again,  in  the  special  units,  the  present  value  of  r  is  11  '454, 
and  that  of  n  is  0-704,  whence  x  is  3-384  ;  also  h  =  4-088.  It  is  plain  that  for 
this  value  of  h  the  biquadratic /(.r)  =  0  has  two  real  roots.  Theksser  of  these, 
xi,  makes  Fa  maximum,  and  the  greater,  x2,  makes  Fa  minimum.  Again, 
f{x)  is  positive  for  values  of  x  between  0  and  xx,  negative  for  those  between  xi 
and  xo,  and  positive  for  those  greater  than  x2.     As  x  is  positive  throughout, 

when  f(x)  is  positive  we  have  — ,>y,  i.e.  «>»;  and  —  <  y,  i.  e.  w  <  w, 

when  f(x)  is  negative.  At  present  f(x)  is  negative,  and  therefore  the  present 
state  of  things  corresponds  to  a  value  of  x  which  lies  between  X\  and  xi. 

We  can  now  determine  the  effects  of  tidal  friction.  Since  the  friction 
resulting  from  the  lunar  tides  constantly  diminishes  the  sensible  or  mechanical 
energy  of  the  Earth-moon  system,  Y  must  continually  decrease  (Art.  282;. 
Hence,  as  at  present  /  (x)  is  negative,  x  must  increase  and  y  decrease  until  T 
reaches  its  minimum,  after  which  the  whole  system  will  move  as  if  rigidly 
connected.  .   .  , 

It  appears  accordingly  that  the  friction  caused  by  the  lunar  tides  diminishes 
the  angular  velocity  of  the  Earth,  i.  e.  increases  the  length  of  the  day,  and  at 
the  same  time  increases  the  Moon's  distance  and  the  length  of  the  month ._  lhis 
process  must  go  on  till  the  day  and  month  are  of  equal  length,  after  which  the 
lunar  tides  will  cease.  If  at  any  past  period  the  Moon  moved  as  if  rigidly  con- 
nected with  the  Earth,  this  must  have  been  when  F  was  a  maximum.  Such  a 
state  of  things  was  dynamically  unstable,  for  the  least  disturbance  of  the  rigidity 
of  the  motion  would'have  produced  tides  whose  friction  would  have  diminished 
the  energy,  and  caused  the  system  to  depart  farther  from  the  configuration  oi 
maximum  energy.  The  departure  from  this  configuration  might  have  taken 
place  in  two  ways,  according  as  the  Moon  approached  the  Earth  or  receded 
from  it.  If  the  former  event  had  occurred,  the  Moon's  angular  velocity  in  its 
orbit  would  have  become  greater  than  the  angular  velocity  of  the  Earth's  rotation, 
and  the  Moon  must  ultimately  have  fallen  upon  the  Earth,  as  x  must  have  de- 
creased continually  along  with  F  If  on  the  other  hand  the  Moon  had  receded 
the  present  state  of  things  would  have  been  reached.  The  value  oi  x  which 
makes  F  a  minimum  lies  between  4-073  and  4-074,  and  the  corresponding 
value  of  n  lies  between  0-015  and  0-014.  The  ratios  of  the  present  value  of  n  to 
these  two  values  are  46-9  and  50-2.  The  present  investigation  would  therefore 
lead  us  to  conclude  that,  when  the  lunar  tides  cease  and  the  day  and  month 
become  equal,  the  length  of  the  day  will  be  between  46  and  50  times  its  present 
length. 


410  Energy. 

Examples  11,  12,  13,  and  Example  3,  Art.  213,  are  taken  from  a  Paper  by 
Professor  G.  H.  Darwin,  first  published  in  the  Proceedings  of  the  Royal  Society 
for  1879,  and  subsequently,  with  some  alterations,  in  Thomson  and  Tait's 
Natural  Philosophy,  Part  ii.  In  this  Paper  Mr.  Darwin  gives  diagrams  of  the 
curves  represented  by  the  equations 

V  =  2Y-K=F  {x),     &f  =  l,     h  =  x  +  y, 

by  means  of  which  the  results  arrived  at  are  exhibited  to  the  eye. 

14.  A  great  number  of  smooth  perfectly  elastic  particles  are  moving  with 
great  velocity  in  various  directions  within  a  rectangular  parallelepiped,  two  of 
whose  opposite  faces  are  large  compared  with  the  others.  If  one  of  these  faces 
be  movable,  determine  the  force  required  to  keep  it  steady. 

Let  u  be  the  velocity  of  one  of  the  particles  whose  mass  is  m,  and  <p  the  angle 
which  the  direction  of  its  motion  makes  with  the  normal  to  the  face.  Before 
striking  the  face  the  particle  has  a  normal  velocity  u  cos  <p,  and  after  the  shock 
it  acquires  an  equal  normal  velocity  in  the  opposite  direction.  The  momentum 
communicated  to  the  face  is  therefore  2mu  cos  $.  Having  reached  the  opposite 
face,  the  particle  rebounds  and  strikes  the  movable  face  again ;  the  interval  of 

.2a 
time  between  two  successive  shocks  against  the  movable  face  being  — ■ — -, 

11  COS  (p 

where  a  is  the  perpendicular  distance  between  the  faces.     The  whole  momen- 
tum communicated  to  the  movable  force  by  the  particle  m  in  the  time  0  is 

therefore  — 0,  and  the  whole  momentum  M  communicated  by  all  the 

a 

0 
particles  in  the  same  time  is  -  "2,mu~  cos-0. 

In  order  to  determine  the  value  of  ~2>nu-  cos2</>,  describe  a  sphere  of  unit 
radius,  and  draw  from  its  centre  lines  parallel  to  the  directions  of  motion  of  the 
various  particles  at  the  beginning  of  the  interval  of  time  0.  Since  the  number 
of  particles  is  very  great  and  the  direction  of  the  motion  of  any  one  undeter- 
mined, we  may  assume  that  the  energy  of  those  particles  whose  directions  of 
motion  make  an  angle  <p  with  a  fixed  direction  is  to  the  total  energy  of  the  system 
as  that  portion  of  the  surface  of  the  sphere  comprised  between  the  cones  whose 
semi-angles  are  <J>  and  $  +  clef)  is  to  the  whole  surface.  If  The  the  total  energy 
of  the  moving  particles,  we  have  then 

fTT  2 

2»*w-  cos2<J>  =  T      cos2tf>  sin  <pd<j>  =  -  T. 

2  TQ 

Hence  M  = .     Now  the  required  force  J1  must  be  such  as  to  communi- 

3  a 

cate  to  the  movable  face  the  momentum  M  in  the  time  0,  and  therefore 

2  Td  2  T 
F6  =  M=-—,  arF=--. 

3  a  3  a 

15.  A  number  of  particles  move  as  in  the  last  example  ;    determine  the 
pressure  which  they  exert  on  the  unit  of  area. 


Examples.  411 

If  S  be  the  area  of  the  movable  face  in  the  last  example,  and  p  the  pressure 

2  T 

of  the  particles  on  the  unit  of  area,  pS=  F=  -  — .    Hence,  if  v  be  the  volume  of 

3  a 
2 

the  parallelepiped,  pv  =  -  T. 

The  results  obtained  in  Ex.  14  and  15  are  made  use  of  to  explain  the  pressure 
which  a  gas  exerts  against  its  envelope.  The  mode  of  investigation  employed 
is  due  to  Clausius. 

16.  Determine  the  mean  kinetic  energy  of  any  system  in  stationary  motion. 

A  system  is  said  to  be  in  stationary  motion  when  the  coordinates  and  the 
velocities  of  its  various  points  fluctuate  within  determinate  finite  limits. 

If  we  integrate  x2dt  by  parts,  we  get  $cb-dt  =  xx-  jxxdt ;  and  similar  equa- 
tions may  be  obtained  corresponding  to  the  other  coordinates.  Again,  supposing 
each  point  of  the  system  to  be  free,  we  have  mx  =  X.     Hence,  if  0  =  t\  —  to, 

1  f'i  1 

Tclt  =  —  ^,m{x\xi  +  yiyi  +  z\i\  -  (x0x0  +  y0y0  +  z0z0)} 
6  Jt0  Zv 

-^  2^(Xx+Y!/  +  Zz)dt. 


If  6  be  made  sufficiently  large,  the  first  term  on  the  right-hand  side  of  this 
equation  may  be  neglected,  and  we  find  that  the  mean  value  of  T  is  equal  to 
the  mean  value  of 

-%2{Xv+  Yy  +  Zz). 

This  latter  quantity  is  termed  by  Clausius  the  virial  of  the  system.  Hence,  the 
mean  kinetic  energy  is  equal  to  the  virial.  This  theorem,  and  its  application 
given  in  Ex.  17,  18,  are  due  to  Clausius,  whose  investigation  will  be  found  in 
the  Philosophical  Magazine  for  August,  1870. 

17.  If  n  be  the  virial  of  a  system  which  is  acted  on  by  no  external  forces 
except  a  uniform  pressure  on  its  surface,  prove  that 

n  =  §pv--\  XS.r<p(r)dt, 

where  r  is  the  distance  between  any  two  points  of  the  system,  v  its  volume,  and 
p  the  pressure  which  it  exerts  upon  the  unit  area  of  the  surface  of  the  surround- 
ing medium  or  envelope. 

If  r  be  the  distance  between  two  particles  of  the  system  whose  coordinates 
are  x,  y,  z;  x,  y',  z',  the  portion  of  IT  due  to  the  mutual  action  of  these  particles 
is  the  mean  value  of  an  expression  of  the  form 

or     -r<p(r).     (Art.  283.) 


412  The  General  Equations  of  Dynamics. 

Again,  if  dS  be  an  element  of  the  bounding  surface  of  the  system  the  di- 
rection cosines  of  whose  normal  are  I,  m,  n,  the  part  of  IT  due  to  the  external 
pressure  is 

\P  jj  {xl  +  ym  +  zn)dS  =  lp{  ^xdydz  +  JJ>  dzdx  +  jjzdxdy}  =  %pv. 
1  fix 

Hence  n  =  f  pv  —    „  2r<£  (r)  dt. 

6Jt0 

18.  Determine  the  pressure  of  a  gas  in  terms  of  its  volume  and  the  mean 
kinetic  energy  of  translation  of  its  molecules. 

A  gas  may  be  regarded  as  composed  of  a  number  of  molecules  which  exert 
no  action  on  each  other  except  when  in  contact.  If  the  gas  be  enclosed  in  an 
envelope,  and  its  condition  remain  unaltered,  its  molecules  must  be  in  stationary 
motion.  Hence,  if  T'  be  the  mean  value  of  that  part  of  the  kinetic  energy 
which  results  from  the  velocities  of  the  centres  of  inertia  of  the  molecules,  and 
n  the  corresponding  virial,  we  have  T'  =  U;  but  n  =  \pv  (Ex.  17),  since  the 
time  during  which  a  particle  is  in  contact  with  other  particles  is  negligible 
compared  with  the  interval  between  two  such  contacts,  and  therefore  the  other 
term  of  n  may  in  this  case  be  neglected.     Accordingly  pv  —  §T\ 


Section  II. — The  General  Equations  of  Dynamics. 

289.   General  Equations  of  Motion  for  any  System. 

— The  general  equations  of  motion  for  any  system  are  ob- 
tained in  precisely  the  same  manner  as  the  general  equations 
of  equilibrium. 

If  F=  0,  G  =  0,  H=  0,  &c,  are  the  equations  of  condition 
representing  the  connexions  and  constraints,  we  have 

dF-       dF„        dF.        dF„ 

—  6xx  +  — -  tii/i  +  —£%  +  -—  &r2  +  &c.  =  0. 
axi  dyx  dzx  dx> 

^  &*  +  &c.  =  0,  ~^xx  +  &o.  =  0,  &o. 
dxx  dxx 

Multiply  the  first  by  A,  the  second  by  /u,  the  third  by  v, 
&c,  and  add  to  D'Alembert's  Equation,  and  we  obtain 

v  d2xx     .dF        dG        dH     g    \.        -M        .    /1N 

Xi-nhW  +  X^+^aVL+Va^  +  &C')^  +  &G'  =  °'   W 

If  there  be  n  equations  of  condition  we  can  assign  to 
A,  ju,  v,  &c,  such  values  as  to  make  the  coefficients  of  the  first 
n  displacements  in  the  above  equation  vanish.     By  means  of 


General  Equations  of  Motion. 


413 


these  displacements  we  can  satisfy  the  n  equations  $F  =  0, 
S6r  =  0,  &c.  The  remaining*  displacements  are  then  entirely 
unrestricted,  and  their  coefficients  in  (1)  must  therefore  be 
each  zero,  and  we  have  for  the  equations  of  motion 


ffixx     _      .dF        clG        dE     _     "| 

mi  7F=Xl  +  Xd7  +  fXd7  +  vaV  +  &c' 

(It  tlXx  LlJLi  tlJbi 

dh,x     _     A  dF        dG        dH 
dt2  dyx        dyx        dyx 


L  -r^r  =  Zx  +  X 


i 

dF 


1  dt 


dG        dE     p 
dzx     r  d%x         dzx 


m2 


d2x2 

w 

&c, 


_     .dF       dG        dH 

X2  +  A  —  +  fl-r—   +V-j-    +  &C. 

dx%        dx2         dXi 


&c, 


&c. 


(2) 


From  these  equations  we  can  obtain  the  Equation  of 
Energy,  if  we  multiply  the  first  by  dxXi  the  second  by  dyx, 
&c,  and  add.     In  this  manner  we  obtain 

(dF  dF  \ 

dT  =  2  {Xdx  +  Ydy  +  Zdz)  +  X  f  —  dxx  +  —  dyx  +  &c.  J  +  &c. 

Now,  if  the  equation  F  =  0  involve  only  the  coordinates 
of  the  various  points, 

dF  _        dF  -  7Z:7     n 

— -  dxx  +  —  c/yi  +  &c.  =  dF  =  05 
flfoi  dyx 

and  the  condition  expressed  by  the  equation  F  =  0  has  no 
effect  on  the  kinetic  energy. 

This  result  was  obtained  from  first  principles  in  Art.  284, 
and  by  its  means  the  Equation  of  Virtual  Velocities  in  its 
usual  form  was  deduced  from  the  Equation  of  Energy. 

290.  Equation  of  Energy  when  Equations  of  Con- 
dition Involve   the  Time  Explicitly. — If  the  equation 


414  The  General  Equations  of  Dynamics. 

F=0  involve  the  time  explicitly,  the  work  done  in  any  actual 
motion  of  the  system  by  the  forces  capable  of  replacing  the 
condition  F  =  0  need  not  be  zero.  In  a  virtual  displacement 
the  work  done  by  these  forces  must  still  be  zero,  because  in 
such  a  displacement  no  lapse  of  time  is  supposed  to  take 
place.  So  far,  therefore,  as  the  equation  of  virtual  velocities 
is  concerned,  t  must  be  considered  constant  in  the  equation 
F=  0,  and  as  in  Art.  200  the  virtual  displacements  must 
fulfil  the  condition 

dF  rfF  dFs        dF. 

-t-COHi  +  -r-Oth  +  —  OSi  +  -7—  bx2  +  &c.  =  0. 
dxi  ay  1  dzv  dxt 

The  actual  displacements  on  the  other  hand  fulfil  the 
condition 

dF.       dF  ,       dF         dF  j        _        (dF\  M      . 
— -  dxi  +  -7-  di/i  +  -—  dzi  +  —-  dx%  +  &c.  +    — -  )dt  =  0. 
dxx  diji  dzx  dxo  \dt  J 

Hence  in  this  case  the  Equation  of  Energy  becomes 
dT  =  2  (Xdx  +  Ydy  +  Zdz)  -  \  f^\  dt-  ^  {^\  dt  -  &c.     (3) 

291.  Similar  Mechanical  Systems. — Two  systems  are 
geometrically  similar  when  each  line  of  the  one  is  equal  to 
the  corresponding  line  of  the  other  multiplied  by  the  same 
constant. 

Similar  Mechanical  systems  are  not  only  geometrically 
similar,  but  have  also  a  similar  distribution  of  mass,  and  a 
similar  distribution  of  force,  and  work  in  a  similar  manner ; 
i.  e.  each  mass  of  the  one  is  equal  to  the  corresponding  mass 
of  the  other  multiplied  by  a  constant,  each  force  of  the  one 
is  equal  to  the  corresponding  force  of  the  other  multiplied  by 
a  constant ;  and  the  systems  are  always  geometrically  similar 
at  instants  of  time  whose  intervals  from  two  fixed  epochs  are 
in  a  constant  ratio. 

Let  x  be  a  coordinate  of  a  point  in  the  first  system,  m  a 
mass,  X  a  force,  and  t  an  interval  of  time ;  and  x',  m\  X',  t' 


Generalized  Coordinates.  415 

the  corresponding  quantities  for  the  second  system  ;  we  have 
then  the  equations  %'  =  Ix,  ml  =  \xm,  X'  =  AX,  t'  =  vt. 

Hence,  Sm'^S*  +  JfW+  ^§*  J 

and       2  (X'&u'  +  F'S/  +  Z'&O  =  A/s  (x^  +  F^/  +  zgs)« 

In  order,  therefore,  that  D'Alembert's  equation  should 
hold  good  for  each  system,  we  must  have  fil  =  \v~. 

This  equation  of  condition  may  be  put  into  another  form 

by  expressing  v  in  terms  of  the  ratio  of  the  corresponding 

velocities  in  the  two  systems.     If  we  denote  this  ratio  by  a, 

dx         dx   -    .     ,      dx       I  dx     ,,       «       ,  j 

we  have  —7  =  a  — ,  but  also,  — -  =  -  —  ;  therefore  I  =  av,  and 
do  do  do        v  do 

the  equation  of  condition  becomes  XI  =  /ma*. 

If,  as  is  generally  the  case,  gravity  be  one  of  the  moving 

forces  in  both  systems,  we  must  have  A  =  ^  ;  hence  a2  =  /,  or 

the  velocity  in  each  system   must  be  proportional  to   the 

square  root  of  its  linear  dimensions. 

292.  Generalized  Coordinates. — If  a  system  have  n 
degrees  of  freedom  its  position  is  completely  determined  at 
each  instant  by  the  values  of  n  independent  variables,  which 
may  be  termed  coordinates,  and  be  denoted  by  £1,  £2,  ?3, . . .  £». 

The  Cartesian  coordinates  x,  ?/,  z  of  any  point  of  the  system 
are  expressible  in  terms  of  these  new  coordinates,  and  are 
therefore  functions  of  the  n  variables  £1,  ?2,  &c.,  these  latter 
being  functions  of  the  time. 

If  we  differentiate  the  equation  x  =  /(?!,  £2,  £3>  •  •  «  ?n) 
with  respect  to  the  time  we  obtain 

,  CtX     A.  CIX     L,  11 X      y.  f  A\ 

d£x        a£t  d%n 

This  equation  shows  that  x  is  a  function  of  the  velocities  fi, 


416  The  Genera!  Equations  of  Dynamics. 

&c,  and  of  the  coordinates  ?l5  &c,  and  is  linear  with  respect 
to  the  velocities.     From  (4)  we  have 

dx       dx       dx      dx     0  /Jrx 

*"£  -S-3&*8-  (5) 

dx 
Again,  if  we  differentiate  —, ■?  with  respect  to  t,  we  get 

dgi 

d    dx      d2x  £         d2x     £  d2x    £ 

dttiZ  =  W?        dfd&  d^a%Jn ; 

but  by  (4)  this  is  the  expression  for  the  partial  differential 

coefficient  — ■ .     Hence  we  have 

d  dx       dx       d  dx       dx    .  f 

dtWi^dZi'    dtd&  =  dQ  {) 

Any  set  of  n  independent  variables  which  completely  deter- 
mine the  position  of  a  system  may  be  taken  as  the  generalized 
coordinates  of  the  system.  The  number  of  these  coordinates 
is  fixed,  bnt  the  actual  coordinates  are  in  general  to  a  great 
extent  arbitrary. 

293.  Kinetic Energy  and  Generalized  Coordinates. 

— The  kinetic  energy  T  of  any  system  in  motion  is  given  by 
the  equation  2T  =  Hm(x2  +  if  +  z~)  If  we  substitute  for  x, 
if  &c,  their  values  given  by  (4)  and  the  corresponding  equa- 
tions, we  obtain  a  homogeneous  quadratic  function  of  the  n 
velocities  f  i,  §2,  •  •  .  ?»,  the  coefficients  of  £ i2,  ji  ?2,  &c., 
being  functions  of  the  coordinates  £i>  £2,  &c,  and  of  the  con- 
stants of  the  system.  If  we  denote  these  coefficients  by  &i, 
23£i2,  &c,  we  have  the  equation 

2T  =  Hx&  +  %&  +  &c.  +  23kf  if,  +  23£j£i£.  +  &c.     (7) 

294.    Equations    of  Motion    for    Impulses.  —  If  a 

system  start  from  rest  under  the  action  of  any  set  of  impulses 
X,  F,  Z,  &c,  the  initial  velocities  are  determined  from 
D'Alembert's  equation  by  equating  to  zero  the  coefficient  of 


Equations  of  Motion  for  Impulses.  417 

each  independent  variation.  Now,  if  ?i,  £>,  &c.  be  the 
generalized  coordinates, 

where  2£i,  S£2,  &c.  are  independent  arbitrary  variations. 
Hence,  substituting  for  §#,  Sy,  &c.  in  D'Alembert's  equation, 
we  obtain  as  the  equation  of  motion  corresponding  to  the 
variation  S£i, 

(  .  dx       .  dy       .  dz\     „  /  _  <rfa      ^ f/v       „  dz  \ 

The  left-hand  member  of  this  equation  becomes,  if  we  sub- 
sttiute  for  -r^r,     -jf?',  &c.  their  values  given  by  (5), 


2w 

/  .  dx       .  f/y 

+  s  -T-     or 
diJ 

Hence,  if  we  put 

s(-x 

at,!        r/gi 

&c, 

we 

have  for 

a  syst 

em  starting  from  rest 

dT 

=  ??*. 

(8) 

In  these  equations  T  is  supposed  to  be  given  by  (7),  and 
2Ji,  &c,  are  the  generalized  resultant  components  of  the 
impulses  tending  to  alter  £,  &c. 

It  follows  from  what  precedes  that,  for  a  system  starting 
from  rest,  D'Alembert's  equation 

Sf»  {xdx  +  yfy  +  z&s)  =  2  (Xfe  +  PSy  +  ZSs) 
becomes  by  transformation  of  coordinates 

^?  S5i  +  ^?  %  +  &c.  =  g,  %  +  £2  Sga  +  &c. 

2E 


418  The  General  Equations  of  Dynamics. 

This  equation  may  be  written  in  the  form 

2(^85)=  s  CK  S2-).  (9) 

If  a  system  be  in  motion,  and  ^>b  jp2,  &c.  be  the  general- 
ized impulse  components  which  would  give  its  actual  motion 
to  the  system  starting  from  rest,  these  impulses  pl9  &c.  are 
determined  by  the  equations 

dT  dT  dT 

Pl  =  l£>       Pi  =  IF' Pn  =  7T'  (10) 

o?gi  d&  d% 

dT     dT 
The  differential  coefficients  -r-,     — -,  &c.  are  the  gene- 

ralized  components  of  momentum  of  the  moving  system. 

If  a  system  in  motion  be  acted  on  by  any  set  of  im- 
pulses whose  generalized  components  are  Hi,  S2,  &c,  the 
changes  of  velocity  produced  by  these  impulses  are  given 
by  the  equations 

dT     (dT\  dT     (dTV     „     .  .„. 

fdTY 
where  (  — -  J ,    &c.   correspond  to   the   instant   immediately 

dT 
before,  and  — =-,  &c.  to  that  immediately  after  the  action  of 

the  impulses.     Since  the  values  of  £l5  &c.  remain  unaltered 
during  the  impulse,  equations  (11)  may  by  (7)  be  written 


ft,  (fe  -  £/)  +  £,  (fc  -  £0  .  .  .  .  +  ft.  [tn  -  in  =  9. 
ft*  (f,  -  ?/)  +  ft*  (&  -  60  ■ . . .  +  ft.  (I.  -  £„')  =  E2  I 

&C  &C.  &C.     y 


Kinetic  Energy  and  Components  of  Momentum.         419 

295.  Kinetic  Energy  and  Components  of  Momen- 
tum.— Since  I7  is  a  homogeneous  quadratic  function  of 
?x,  £2,  &c,  we  have,  by  Euler's  Theorem, 

•   dT      .  dT 
2T=%X~  +  f3i±  +  &c.  -*&+!*&.  •  .^»5«.      (13) 
dKi  d& 

This  equation  may  be  written  in  the  abbreviated  form 

2r=SQ4).  (14) 

If  we  suppose  the  same  system  occupying  the  same  posi- 
tion to  have  successively  two  different  motions,  in  the  first  of 
which  the  velocity-components  are  f:,  &c,  and  in  the  second, 
§/,  &c,  and  if  we  put  f/=  £i  +  a1?  &c,  and  express  the  cor- 
responding values  of  T  by  T\  and  Tj? ,  we  have,  by  Taylor's 
Theorem, 

Tj,  =  Ti+Vpa  +  Ta,  i.e.  r^=r|+Sp(f'- |) +  %_|).    (15) 

If  now  we  suppose  a  system  to  start  from  rest  the  values 
of  certain  components  of  velocity  being  prescribed,  and  if 
the  system  be  set  in  motion  by  impulses  such  that  there  are 
no  components  of  impulse  except  those  corresponding  to  the 
prescribed  velocities,  the  initial  kinetic  energy  is  a  minimum. 

Let  £i,  &c,  be  the  velocity-components  of  the  initial 
motion  produced  in  the  manner  described,  then  pl9  &c,  are 
the  impulse-components;  and  if  any  impulse-component pq 
be  not  zero,  the  corresponding  velocity-component  %q  is  pre- 
scribed. Let  us  now  suppose  the  system  to  be  set  in  motion 
in  any  other  way,  the  prescribed  velocity-components  being 
the  same  as  before,  and  let  |/,  &c,  be  the  velocity- components 
of  the  new  initial  motion.  We. have,  then,  2p  (?'-  ?)  =  0, 
since  whenever  £>  is  not  zero,  £'  =  %.  Hence  T%  =  T%  +  ^ (£'-£), 
and  therefore  T*  is  a  minimum. 

This  is  Thomson's  Theorem,  Art.  199. 

Again,  if  fi,  &c,  pl9  &c.  be  the  components  of  velocity 
and  momentum  of  a  system  in  any  given  position,  and  ?/, 
&c,  p{9  &c.  the  corresponding  quantities  for  a  different 
motion  of  the  same  system  in  the  same  position,  we  have 

s  (ptr>  =  s  (/?).  (i6) 

2E  2 


420  The  General  Equations  of  Dynamics. 

The  truth  of  this  equation  appears  readily  by  substituting 
fi  +  £/,  &c.  for  |i,  &c.  in  T%,  and  equating  the  two  expres- 
sions which  by  Taylor's  Theorem  can  thence  be  obtained. 

296.  Energy  of  Initial  motion. — If  we  substitute  %vdt 
for  S£i,  ^dt  for  S£2,  &c.  in  (9),  we  obtain  for  the  initial 
energy  T  of  a  system  starting  from  rest  the  equation 

2r=S(S?).  (17) 

Let  us  now  suppose  that  on  a  system  having  £x,  &c.  as 
its  generalized  coordinates  constraints  are  imposed  capable  of 
being  expressed  as  in  Art.  284  by  equations  connecting  the 
coordinates  &,  £2, . . . .  gn.  The  coordinates  are  then  no  longer 
independent,  and  if  the  system  be  set  in  motion  by  impulses 
5Efi,  &c,  equations  (8)  no  longer  hold  good,  but  (9)  and  (17) 
remain  valid,  f^  &o.  being  the  velocity- components  of  the 
actual  motion.  Also  T  is  the  same  function  of  fi,  &c.  as 
it  was  before  the  imposition  of  the  constraints,  the  only 
difference  being  that  certain  relations  hold  good  in  the 
constrained   motion   connecting    these   velocity-components. 

In  order  to  compare  the  initial  kinetic  energies  of  the 
system  in  the  unconstrained  and  constrained  motion,  let  &, 
&c.  be  the  velocity-components  corresponding  to  the  former, 
and  f/,  &c.  those  corresponding  to  the  latter,  then  by  (17) 
we^have 

22>  =  2(Er)  =  S(i>|')>  also  2Tj:  =  2pt 
Substituting  in  (15),  we  obtain 

This  proves  Bertrand's  Theorem,  Art.  199. 

297.  Iiagrange's  Equations  of  Motion. — We  saw  in 

Art.  294  that 

/.  dx       .  dy       .  dz\      dT  „_, 

V   d^i        d£L        d&J      d%x'  V     ' 


Lagrange  s  Equations  of  Motion. 


421 


if  we  differentiate  each  side  of  this  equation  with  respect  to 
the  time,  and  substitute  for  —  -r=-,    &c.  their  values   given 

Clt    usi 

by  (6),  we  obtain 


dx      ..  di/ 


d 
It 


'z\  f .  dx      .  dy      .  dz 

F    +  2m  [x  -p  +  y  -p£  +  z  -^ 
t,ij  \    dt.x         d%x        dt,, 


d_dT  t 
dtd%} 


dT 


t     .  „     / .  dx       .  dt)       .  dz\  .      -,   .   -■ 

but  2m   x  —r  +  z-k?  +  z  -tt)  is  plainly    , 

V  d&    J  d&        dlx)       *        J  dlx 

hence  we  have 

„     /    dx      ..  dy      ..  dz  \      d  dT     dT 

2w  [x  -rr  +  y  -£■  +  z  -?    =  — ■  -3-  -  -r— .      (20) 

V  rf&     J  d&        dZj     dtdl     dlx       v     ' 

Now  in  D'Alembert's  equation  for  continuous  forces  the 
coefficient  of  the  independent  variation  §&  is, 


dx       .  dy         dz 
dlx     J  d&        dl, 


d&  d&         dZi 


Hence,  if  we  put 


_  dx       T_  dy       „  dz\      _     _ 

we  have,  as  the  equations  of  motion  of  any  system, 

d_  (IT  _  dT  _ 
.  dt  dtx     dti  ~      * 


d_  dT_  dT 
dt  4*     * 


>• 


(21) 


dt    dt       dln    '        "    J 

The  work  which  would  be  done  by  the  forces  of  the 
system  against  the  displacement  §£i  is  -  Si  S?i,  accordingly 
35?i,  &c.  are  the  generalized  components  of  force  tending  to 
alter  the  coordinates  &,  &o.     It  is  to  be  observed  that  the 


422 


The  General  Equations  of  Dynamics. 


forces  X,  F,  Z,  &c.  are  not  equivalent  to  the  forces  Hi,  S2, 
&c,  unless  the  variations  $x,  By,  dz,  &c,  and  the  correspond- 
ing variations  Sgi,  S£2,  &c,  result  by  orthogonal  projection 
from  the  same  possible  displacements  of  the  system. 
For  a  conservative  system  equations  (21)  become 


d_  dT 
It  d%x 

d_  dT 
It  d& 


dT     dV_ 
d&  +  & 


r    dv 

dh  +  dl, 


d_  dT 

dt  d%n 


dT     dV 

+  — 

d%n       d£n 


u,     ^_^_+^_=0 


1 


^  +  ^-=o   y 


J 


(22) 


Equations  (21)  and  (22)  were  first  given  by  Lagrange, 
and  are  known  as  Lagrange's  equations  of  Motion  in 
Generalized  Coordinates. 

The  proof  given  above  for  Lagrange's  equations  holds 
good  even  though  the  time  appear  explicitly  in  the  equations 
which  determine  the  Cartesian  in  terms  of  the  Generalized 
coordinates.     In  this  case  x  =/(?i,  ?2, 5n,  t),  &c-  *  taen 

contains  the  additional  term  — ;  but  the  equations 

dt 


dx       dx  d   dx  _  dx 

4i =  a%'   G''  It  S& ~ 7&   c'' 

are  still  true,  and  therefore  the  proof  of  Lagrange's  equations 
remains  valid. 

If  we  put  L  =  T  -  V,  the  function  L  is  the  difference 
between  the  kinetic  and  potential  energies  of  the  system, 
and  is  called  Lagrange's  Function. 

Equations  (22)  may  now  be  written  in  the  form 


d  dL     dL 

dt  d%x      d%i 


^  _  £5  =  0.  (23) 

dt  d%n      d%„ 


Effect  of  Constraints.  423 

This  form  of  the  equation  of  motion  is  likewise  due  to 
Lagrange. 

298.  Deduction  of  the  Equation  of  Energy. — If  we 

multiply  the  first  of  equations  (22)  by  fi,  the  second  by  £>, 
&c,  and  add,  we  get 

±ld    dT     dT\     ^{j.dV\     n  ,nA, 

2?  —    -r 1  +  2    ?—     =0.  24 

•  dT 
Now  2T=2?-r-; 

and  therefore     2  —  =  2   ?  —   —r  +  ?  -f  > 
hence 

"•ej-s)-f-«(«f*'S)-f'« 


rfT      ^/^rfT      *dT 

since  =  2    §  — -  +  §  — r 

Substituting  in  (24),  we  obtain 

(IT     dV  _ 
dt  +  dt    "    ' 

hence  we  have 

T+V=E.  (26) 

299.  Effect  of  Constraints. — If  a  system  having  n 
degrees  of  freedom  be  subjected  to  additional  constraints 
capable  of  being  expressed  by  q  equations  connecting  the 
coordinates  of  the  form  F=  0,  G  =  0,  &c,  we  may  either 
select  a  new  system  of  n  -  q  generalized  coordinates,  or  else 


424  The  General  Equations  of  Dynamics. 

retain  the  old  system,  and  proceed  according  to  the  method 
of  Art.  289.     Equations  (22)  would  then  become 

d    all     dT     dV      .dF         dG      _        x 
dt  dll~df1  +  d^  =  Xdfl+fXd^  +  &G' 

d   dT     dT     dV      .dF        dG      a 

d_'dT_dT     dV  _  .dF        dG_ 

dt  d%n     d%n      d£n  d%n     **  d%n 

In  the  case  of  impulses  we  may  proceed  in  a  similar 
manner,  and  still  make  use  of  equations  (8)  or  (11),  provided 
we  introduce  additional  terms  into  S?i,  &c.  representing  the 
impulses  by  which  the  constraints  may  be  replaced.  It  is 
plain  that  both  in  the  case  of  continuous  and  also  in  that  of 
impulsive  forces  the  terms  in  Lagrange's  equations  repre- 
senting the  action  of  the  constraints  disappear  from  the 
equation  of  energy. 


Examples. 

1.  Determine  in  polar  coordinates  the  equations  of  motion  of  a  particle 
which  moves  freely  in  a  fixed  plane. 

Here     T=\m  (r2  +  r262),  whence 

d  dT      dT        ..  .,     d  dT     dT         d  ,     . 

- — —  — —  =  mr  —  mr9*,    —  -p-  -  — -  =  m-r  (r29), 
dt  dr       dr  '   dt  dd       dd  dtK      h 

and  the  equations  of  motion  are  the  same  as  those  which  would  be  given 
by  (11)  and  (12),  Art.  28. 

2.  Determine  in  polar  co-ordinates  the  general  equations  of  motion  of  a 
free  particle. 

Here  T  =  §  m  { r2  +  r2  (02  +  sin2  9<p2) } , 

and  the  equations  of  motion  are 

in  {r  -  r  {92  +  sin2 9(p2) }  =  £, 

i  I  t-  {r2  9)  -  r2  sin  9  cos  d<j>2  J  =  Pr,     m  -  {r2  sin2 9$)  =  Qr  sin  9, 


Examples.  425 

where  i?,  P,  and  Q  are  the  components  of  the  force  acting  on  the  particle,  along 
the  radius  vector  from  the  origin,  perpendicular  to  the  radius  vector  in  the 
meridian  of  the  particle,  and  at  right  angles  to  these  two  directions. 

3.  Prove  Euler's  equations  for  a  body  having  a  fixed  point. 

The  body-axes  being  the  principal  axes  at  the  fixed  point,  the  expression  for 
Tin  terms  of  0,  <p,  i//  is  given  in  Ex.  5,  Art.  288.     Hence 

—  {  C<p  +  dp  cos  0 }  -  {A  -  B)  sin  <p  cos  <p  (02  -  i£2  sin20) 

+  {A  -  J5)sin0  (cos20  -  sin2c|>)  0^  =  *. 

If  we  substitute  u>z  for  <p  +  ^  cos  0  by  (12),  Art.  258,  and    then   make 
0  =  <p  =  1 7T,  we  have 

C  d-p  -  (A  -  B)  co!  W3  =  *  =  N.     (Ex.  4,  Art.  288.) 

4.  Generalize  Euler's  equations  for  the  case  in  which  the  body-axes  are  not 
principal  axes. 

In  this  case  T  is  a  quadratic  function  of  &n,   coo,   cos,    with   constant  co- 
efficients (Art.  263).     Hence,  by  Ex.  5,  Art.  260, 


dT     dT  dcci      dT  dw2      dT  dm      dT 
d(p       da)\  dcp       dwz  d<p       does  d<p       dv% 

dT     dT  dwx      dT  cfcoo      dT  dw3  _  dT 
d(p      du}\  d(p       dwz  d<p       da)3  dp       dwi 

dT 

2—  T~  wl  '■> 
aoo-2 

and  we  have 

d  (dT\       dTl         dT 

—   (  J    —    C02  + COl  =  4>  = 

dt  \dcoz/        dan          dwo 

=  N. 

5.  Two  particles  m  and  m  are  connected  by  an  inextensible  string  passing 
through  a  smooth  hole  at  the  edge  of  a  smooth  horizontal  table  on  which  m  rests  ; 
determine  the  equations  of  motion  of  the  particles,  and  the  tension  of  the  string. 

Let  r  and  6  be  the  polar  coordinates  of  m  with  respect  to  the  hole  as  origin  ; 
then 

IT  =  (m  +  m)  r3  +  mr2  02, 

and  the  equations  of  motion  are 

d 
(m  +  tri)  r  —  mr  6~  =  -  m'g,     —  {mr2  6)  =  0. 

If  t  be  the  tension  of  the  string,  and  h  the  value  of  mr2  9,  we  have 

mr  —  mrd2  =  -  r  (Ex.  1), 


whence 


mm       (  h2  \ 

=  — : — ;  Iff+TZ)' 
m  +  m    \       mira/ 


6.  A  smooth  particle  descends  the  upper  edge  of  a  thin  vertical  lamina 
which  is  capable  of  sliding  freely  down  a  smooth  inclined  plane  with  which 


426  The  General  Equations  of  Dynamics. 

its  whole  lower  ledge  is  in  contact.  If  the  plane  of  the  lamina  he  perpendicular 
to  the  intersection  of  the  inclined  plane  with  the  horizon,  and  the  particle  and 
lamina  start  from  rest,  determine  their  position  at  any  time. 

Let  x  he  the  distance  at  any  time  of  a  point  in  the  hase  of  the  lamina  from 
its  initial  position,  |  the  distance  which  the  particle  has  moved  along  the  edge  of 
the  lamina,  a  the  angle  which  this  edge  makes  with  the  inclined  plane,  &  the 
inclination  of  the  latter,  m  the  mass  of  the  particle,  and  M  that  of  the  lamina. 

The  kinetic  energy  of  the  lamina  at  any  time  is  ^  Mx2,  and  that  of  the 
particle  is 

\m  { (x  +  £  cos  a)2  +  £2  sin2  a} . 
Hence 

2T=  (M  +  m)  x-  +  m£  +  2mx  £  cos  a. 

Again,  —  V  =  Mgx  sin  /3  +  mg  { (x  sin  £  +  |  sin  (a  +  j8) } , 

and  therefore  the  equations  of  motion  are 

(M  +  m) x  +  m'(  cos  a  =  (M  +  m)  g  sin  j8,     m  (£+  x  cos  a)  =  mg  sin  (a  +  &), 

whence 

(  .  m  sin  a  cos  a  cos  £ )  .    j9  (M  -4-  m)  sin  a  cos  £ 

2  J     \  M+m  sm?-  a      )  a  Jf  +  w  sin2  a 

300.  Ignoration  of  Coordinates. — If  there  be  no  force 
tending  to  alter  one  or  more  of  the  independent  variables 
by  which  the  position  of  a  system  is  defined  ;  if  moreover 
the  expression  for  the  kinetic  energy  of  the  system  does  not 
contain  these  variables,  bnt  only  their  differential  coefficients  ; 
and  if  the  system  start  from  rest ;  then  T  may  be  expressed 
as  a  function  of  the  other  independent  variables  and  their 
differential  coefficients,  and  be  treated  as  if  these  latter  vari- 
ables completely  denned  the  position  of  the  system. 

Let  &  be  one  of  the  independent  variables  satisfying  the 
conditions  supposed ;  then,  as  there  is  no  force  tending  to 
alter  &, 

dT 
0  ;  and  therefore  -r-  =  constant ; 
d\x 

also  as  the  system  starts  from  rest,  and  T  is  a  homogeneous 

quadratic  function  of  f  1,  £2 . . .  tm  this  constant  must  be  zero  ; 

dT 
hence  — -  =  0.     In  like  manner,   if  £>  be  another  variable 

d%x 

dT 
satisfying  the  same  conditions,  we  have  —r-  =  0,  and  so  on. 

dc,^ 


d   dT 

dt  df, 

Ignoration  of  Coordinates.  427 

dT  dT  •     • 

From  the  linear  equations  —r-  =  0,   -r  =  0,  &c,  £1,  £2,  &c,  can 

be  found  in  terms  of  the  remaining  differential  coefficients 
2^,  .  .  .  %n.  Thus  T  becomes  a  function  of  ifg  . . .  £n,  and  of 
their  differential  coefficients,  that  is 

r  =  i?(5fi,??+1,&c.,  ?„&c). 

If  now  we  regard  f?,  £ff+1,  &c.  as  completely  defining  the 
position  of  the  system,  Lagrange's  equations  are 

d    dF     dF_ 

It  d$q  ~  «c  "  S*'  &C* ; 

but  these  equations  are  true,  for 

dF     dT     dT  d&      dT  d&       ' 

—r    =  —7-  +  — r       — -  +  — -     — ~  +  &C, 

rf?f  dit  d&  d%q  di>  diq 

clF  _dT_     dT  dt      f^4i  +  &c. 
d%q      d%q      d%x  d%q      d%%  d£q 

whence,    as      — -  =  0,     — r  =  0,  &c,  we  have 
clF_dT      clF_dT 

dtq         d\q         d%q         ^q 

The  proposition  proved  above  is  given  by  Thomson  and 
Tait  {Natural  Philosophy),  and  is  the  simplest  case  of  what 
they  have  termed  Ignoration  of  Coordinates. 


Examples. 

1 .  A  particle  descends  from  rest  along  one  face  of  a  smooth  triangular  prism 
which  is  supported  by  a  smooth  horizontal  plane.  The  initial  position  of  the 
particle  lies  in  the  vertical  plane  containing  the  centre  of  inertia  of  the  prism 
and  perpendicular  to  its  edge  ;   determine  the  motion. 

Let  x  be  the  horizontal  coordinate,  in  the  vertical  plane  in  which  the  particle 
moves,  of  the  centre  of  inertia  of  the  prism,  M  its  mass,  m  that  of  the  particle, 


428  The  General  Equations  of  Dynamics. 

I  the  distance  it  has  moved  at  any  time  along  the  face  of  the  prism,  and  o  the 
angle  which  this  face  makes  with  the  horizontal  plane  ;  then 

2T=(M+  m)  x2  +  mi?  +  2mx\  cos  a,      V——  mgl  sin  a ; 

and  the  equations  of  motion  are 

(If  +  m)  x  +  mi  cos  a  =  0,     m\  +  mx  cos  a  =  mg  sin  a. 

Hence,  as  the  particle  starts  from  rest, 

(M  +  m)  x  =  ~mi,  cos  a,     (M  +  m  sin2  o)  f  =  (M  +  m)  g  sin  a. 

The  student  will  observe  that  if  T  were  expressed  by  means  of  the  first  of 
these  equations  as  a  function  of  £  alone,  and  treated  as  such,  the  second  equation 
would  be  obtained  directly  as  Lagrange's  equation. 

2.  In  the  preceding  example,  if  the  face  of  the  prism  down  which  the  par- 
ticle descends  be  rough,  determine  the  equations  of  motion. 

The  force  of  friction  tends  merely  to  stop  the  relative  motion  of  the  particle 
and  prism  ;  hence,  F  being  this  force,  F8f=  -  juP5|,  where  P  is  the  perpen- 
dicular pressure  of  the  particle  on  the  face  of  the  prism.  Now  P  =  m(g  cos  a 
+  x  sin  a),  and  therefore  the  equations  of  motion  are 

(M  +  m)  x  +  m\  cos  a  =  0, 

m\  +  mx  cos  a  =  mg  (sin  a  —  ft  cos  a)  —  ixmx  sin  a. 

The  latter  of  these  equations  can  be  reduced  to  the  form 

£  cos  X  +  x  cos  (a  -  A)  =  g  sin  (a  -  A), 
where  tan  x  =  [x. 

3.  A  sphere,  having  no  motion  of  rotation,  and  under  the  action  of  a  force 
passing  through  its  centre  of  inertia,  moves  through  a  liquid  extending  indefi- 
nitely in  all  directions  on  one  side  of  an  infinite  plane  :  the  liquid  being  origi- 
nally at  rest,  and  not  acted  on  by  any  force,  determine  the  form  of  the  equations 
of  motion  of  the  sphere. 

Let  the  origin  be  anywhere  in  the  fixed  plane,  the  axis  of  x  being  at  right 
angles  to  that  plane  ;  and  let  x,  y,  z  be  the  coordinates  of  the  centre  of  the 
sphere  at  any  time,  and  £  a  coordinate  of  any  particle  of  the  liquid,  which  may 
be  defined  as  matter  which  is  incompressible,  devoid  of  resistance  to  change  of 
shape,  and  incapable  of  exercising  any  friction  against  a  surface  with  which  it 
is  in  contact. 

dT 

If  T  be  the  kinetic  energy  of  the  whole  system,  we  have  —p  =    C,   since 

there  is  no  force  acting  on  the  liquid  ;  but  as  the  liquid  was  originally  at 
rest,  and  no  impulse  was  imparted  to  it,  C  =  0.  Hence_  T  is  a  function  of 
x,  y,  z,  x,  y,  z.  Again,  the  motion  of  the  system  at  any  instant  could  be  pro- 
duced from  rest  by  placing  the  sphere  in  its  actual  position,  and  giving  it  an 
impulse  sufficient  to  impart  to  it  its  actual  velocity,  since  the  impulses  which 
should  be  given  to  the  liquid  particles  are  zero  (10),  Art.  294.  Hence,  asthe 
initial  circumstances  are  unaltered  by  changing  the  values  of  y  and  z,  I7  is  a 
function  of  x,  x,  y,  z.    Again,  a  change  in  the  sign  of  y  or  z  can  make  no  change 


Components  of  Momentum  and  Velocities.  429 

in  the  value  of  T,  which  must  therefore  he  of  the  form  ^{Px2  +  Q  (y2  +  i3)}, 
since  the  coefficients  of  xy,  yz,  zx  must  he  zero. 
The  equations  of  motion  are  then 

Qu  +■  -f  *y  =  1,    Qz  +  -^xz  =  ^, 

4.  Prove  that  a  sphere  projected  through  a  liquid  perpendicularly  from  an 
infinite  plane  boundary  is  at  first  accelerated,  and  then  tends  towards  a  con- 
stant velocity.  Show  also  that  if  projected  parallel  to  the  boundary  it  moves  as 
if  it  were  attracted  towards  the  boundary. 

Initial  circumstances  in  Ex.  3  are  altered  in  the  same  manner,  whether  we 
suppose  introduced  into  the  liquid  a  second  bounding-  plane  parallel  to  the 
first,  and  between  it  and  the  sphere,  or  suppose  the  sphere  placed  initially 
nearer  the  original  bounding  plane.  Hence  a  diminution  of  the  initial  value 
of  x  is  equivalent  to  the  introduction  of  additional  geometrical  constraints 
into  the  system.  From  this  it  follows  by  Bertrand's  Theorem,  Art.  296,  that 
if  x'  <  x,  and  P'x'  =  Px,  the  value  of  Px2  must  exceed  that  of  P'x'2,  and  there- 
fore x  <  x,  and  P"  >  P,  or  P  decreases  as  x  increases.  Similar  reasoning  can  be 
applied  to  Q.  If  x  be  infinite,  or  the  liquid  unbounded  in  every  direction,  P 
and  Q  are  constants. 

The  statements  made  in  the  enunciation  of  this  example  follow  then  imme- 
diately from  the  equations  of  Ex.  3,  by  making  X  and  Y  zero. 

Examples  3  and  4  are  taken,  with  some  slight  modifications,  from  Thomson 
and  Tait  {Natural  Philosophy). 

301.   Components  of  Momentum  and  Velocities. — 

Equations  (10),  Art.  293,  enable  us  to  express  the  velocities 
Ei,  &c.  as  linear  functions  of  the  components  of  momentum 
Pi,  &c.  If  these  values  be  substituted  for  Ei?  &c.  in  T,  as 
given  by  equation  (7),  a  new  expression  for  T  is  obtained 
which  is  a  homogeneous  quadratic  function  oiplyp2,  . .  .  pn. 
We  shall  represent  the  two  expressions  for  T  by  T*  and  Tp. 
Equation  (7),  Art.  293  gives  jT>,  and  the  corresponding 
equation  for  Tp  is  of  the  form 

Tp  =  Pnpc  +  P22P22  +  &c.  +  2P12p1p2  +  &c.       (28) 

In  this  equation  Pn,  P22,  &o.  are  functions  of  Ei>  fe>  &o. 
Thus  Tz  and  Tp  are  each  functions  of  Ei,  ?2,  &c. ;  but  these 

coordinates,  so  far  as  they  appear  explicitly,  do  not  enter  in 
the  same  manner  into  the  two  expressions  for  T.     Equation 


430  The  General  Equations  of  Dynamics. 

(14)  gives  an  expression  for  T  which  is  symmetrical  in  £1  and 
pi,  &c,  and  which  becomes  T$  or  Tp  according  as  we  express 
ph  &c.  in  terms  of  %Vl  &c,  or  &,  &c.  in  terms  of  p„  &c. 

If  we  seek  for  —  from  equation  (14)  we  obtain 
dpi 

2-r-=  gi  +  S^  — .  (29) 

>  Again,  if  we  seek  for  —  from  (7)  we  have 

dT      dTdt      dT  df,      o         v    4  ,oftv 

=    — —    +      — ; +     &C.      =     2^    •  (OO) 

dpi      d%x  dpx      f/?2  dpx  dp, 

dT 
Substituting  this  value  for  —  in  (29)  we  get 

ndT      *      dT  dT     t 

2  -—=?!  +  — ,  whence    —  =  \\\ 
dpx  dp'  dpx 

and  as  a  similar  result  holds  good  for  each  component  of 
momentum,  we  have 

dT      g      f^     t  dT      * 

The  partial  differential  coefficients  of  T  with  respect  to 
£l5  &c.  are  different  according  as  T  is  expressed  by  T^  or  Tp. 

dT 
If  we  seek  for  -~   from   equation  (14)    or  (7)   we  must   m 

each  case  regard  &,  &c.  as  functions  of  #,  ^2,  &c. ;  &,  ?2,  &c. 
In  this  way  we  get  from  (14), 


*%-»%**%***-         (32) 


Hamilton' '$  Equations  of  Motion.  431 

and  from  (7) 

alTp      dTi     dTi  dt      dTt  d%% 

—  =  — -  +  -4  —  +  —4  —  +  &c. 

d&       d&       rff,   rff,      d&  rff, 


+  Pi  -Tg-  +  i?2  -r=-  +  &c.  (33) 


iff,     J    d&     " <% 

Hence,  by  (32),  *  =  _f  +  2  _£, 

and  therefore  £  + I  _  q 

We  have  then  the  system  of  equations 

dTp     dT,  clTp     dT,  dTp     dT, 

It  is  plain  that  the  reciprocal  relations  between  compo- 
nents of  velocity  and  momentum  are  analogous  to  the  polar 
properties  of  curves  and  surfaces. 

302.   Hamilton's   Equations  of  Motion. — If  we  put 

Tp  +  V  -  U,  we  obtain  a  function  27" of  Pi,pz,  &c,  £u  ?2,  &&., 
which  represents  the  total  energy  kinetic  and  potential  of  the 
system,  and  whose  value  is  constant.  By  the  employment 
of  U  Lagrange's  equations  of  motion  may  be  expressed  in 
another  very  symmetrical  form  due  to  Hamilton. 

-d     /ian    a  l  nnA     d  dT     dpi  dT     dTp 

By  (10),  Art.  294,  -^  =  g,  and  by  (34)  -  ^  =  -g. 

Hence  Lagrange's  equations  (22)  become 

f  +  ^0,f  +  §  =  o,...%  +  ^-0.      (35) 
dt       d%x  dt       d%i  dt       d%n  v     ' 

Equations  (35)  have  been  termed  The  Equations  of 
Motion  of  a  system  expressed  in  the  Canonical  Form. 


432  The  General  Equations  of  Dynamics. 

It  is  easy  to  see  that  tlie  equations  which  give  the  motion 
of  the  centre  of  inertia  and  the  changes  in  the  moments  of 
momentum  for  any  system  are  particular  cases  of  equa- 
tions (35). 


Examples. 

1 .  In  a  moving  system  the  total  elementary  change  of  momentum  corre- 
sponding to  one  of  the  generalized  coordinates  is  made  up  of  two  parts,  one 
resulting  from  the  forces  acting  on  the  system,  the  other  from  the  previously 

dT 
existing  motion.     Show  that  — -  dt  expresses  the  latter,  |  being  the  generalized 
d\ 

coordinate. 

If  »,  &c.  be  tha  impulses  which  would  give  the  existing  velocities  at  any 
dT  .  (dTV       , 

instant,  —  =  p.     At  the  next  instant  (  —v- )  =p  . 
d£  \«|  / 

From  these  equations  it  appears  that  the  total  elementary  change  of  mo- 
mentum p'  —  p  corresponding  to  £  is 

dT\'     dT  d  dTJ 

at. 


fdT\  '     dT  d_dl 

Ui  /     «*£    or  *  4 

Now,  by  Lagrange's  equations 


d    dT 

,       dT  , 

—   —dt  = 

s.dt  +  —  dt 

dt   dt, 

di 

whence,  as  s.dt  represents  the  change  of  momentum  resulting  from  the  applied 

dT 
forces,   — dt  must  represent  that  due  to  the  previous  motion. 
d\ 

2.  Apply  the  method  of  the  last  example  to  determine  the  components  of  the 
centrifugal  couple  in  the  case  of  a  body  having  a  fixed  point. 

Here  IT  -  Aai2  +  Bootf  +  Cwi1.     If  now  m,  wz,  a>3  be  expressed  in  terms  of 

e,  <j>,  $ ;  e,  <p,  h 

dT  _  dT  dm      dT    daZ      d/T_    dm 
d<p      du)\  d<f>       doo2    d<p       dwz   d<p  ' 

then  when  <p  =  0,  we  have,  Art.  258  and  Ex.  5,  Art.  260, 

dT       I  A        m 

—  =  {A  —  B)  m  a>%. 

d(j> 

3.  If  the  Cartesian  and  generalized  coordinates  be  connected  by  linear 
equations  with  constant  coefficients,  show  that  there  are  no  terms  in  the  equations 
of  motion  resulting  from  the  previous  motion. 


Calculus  of  Variations.  433 

303.  Calculus  of  Variations. — In  the  Calculus  of 
Variations  the  form  of  the  function  which  determines  the 
dependent  variable  y  in  terms  of  the  independent  variable  x 
is  supposed  to  vary,  and  zs  being  the  symbol  of  a  given 
operation  or  set  of  operations,  the  fundamental  problem  of 
the  Calculus  is  to  determine  the  variation  of  zsy. 

If  y  =/(,r),  a  change  whose  magnitude  is  infinitely  small 
in  the  function  /  (x)  must  be  of  the  form  i\p  (x) ,  where  i  is  an 
infinitely  small  constant.  We  have  then  By  =  ty  (x).  In 
consequence  of  y  becoming  f(x)  +  i\p{x),  the  differential  co- 

m  ■     4.  ^  \>  d>[f    \d>^ 

efficient  — -  becomes    — -  +  i  -—■. 
dxn  dxn        dxn 

Hence  we  have  o  -r-=  =  —j-r-  U>b) 

dxn       dxn 


If  Q  = 


i*»£-2i 


the  variation  £12  is  the  change  in  £2  in  consequence  of  y 
changing  from/(.r)  to  f(x)  +  i\p  (a?).  As  the  result  of  this 
change  of  y  the  function  F  becomes  F  +  SF,  where 

sw-dFx  dFdSy  dF     dn$u 

~dyJ+  ~JW\   dx    ""+  d(d»y\    dx»  ' 


\dx) 


and  12  becomes  j  Fdx  +  f  SFdx.    Hence  we  see  that 

$Q  =  d$Fdx  =  fSFdx.  (37) 

In  the  case  of  a  definite  integral  whose  limits  are 
variable  the  complete  variation  is  the  sum  of  two  parts,  one 
resulting  from  the  variation  of  the  limits,  the  other  from  the 
variation  of  the  expression  under  the  integral  sign.     Hence 

if  12  =        Fdx,  and  if  D£2  be  the  complete  variation  of  12,  we 

X 


have  DQ  =  F"dx"  -  F'dx' + 


SFdx.  (38) 


In  general  the  complete  variation  Du  of  a  dependent 
variable  u  is  the  sum  of  two  parts,  one  resulting  from  a 
change   of  the   independent   variable  x,  the  other  from    a 

2F 


434  The  General  Equations  of  Dynamics. 

change  in  the  form  of  the  relation  connecting  u  with  x.  In 
the  Calculus  of  Variations  the  symbol  8  is  restricted  to  varia- 
tions of  the  latter  kind.     Hence,  in  general, 

Da  =  C^dx  +  hi.  (39) 

ax 


Examples. 

1.  A  particle  under  the  action  of  gravity  is  constrained  to  move  from  one 
given  point  A  to  another  B  along  a  smooth  plane  curve  ;  determine  the  nature 
of  the  curve  so  that  the  time  of  descent  may  be  the  least  possible. 

The  curve  obviously  lies  in  a  vertical  plane  passing  through  the  points 
A  and  B. 

Let  the  axes  of  x  and  y  be  a  vertical  and  horizontal  line  in  this  plane,  the 
positive  direction  of  x  being  downwards,  and  let  v  be  the  velocity  of  the  particle 
in  any  position,  then,  if  the  origin  be  properly  selected, 

ds 
v2  =  2y x,  and  therefore  dt  =  . 

yflg* 

dy 


ax 


Hence,  if  a  =  j  *  J~^f  dx>    wnere 

we  have  to  determine  y  as  a  function  of  x  so  that  n  may  be  a  minimum,  and 
therefore  5n  =  0  for  all  possible  variations  of  y.     Now 

8n=p      p   ■  «rfc 

hence,  integrating  by  parts,  and  neglecting  the  terms  outside  the  integral  sign , 
since  y\  and  yo  are  given,  and  therefore  5yi  =  dy0  =  0,  we  have 

p    d_t  P  )5ijdx  =  0, 

ho   dx  W'2gx(l+p~)/ 

but  8y  being  arbitrary,  this  equation  cannot  be  true  for  all  values  of  Sy,  except 

^     ,-        *  =0. 

dx  N-lgx  (1  +  pf 

Integrating,  we  have  p1  =  2yc2x  (1  +  i^2). 

1  •  o        dy 

If  we  put        — -„  =  a.  and  p  =  tan  0,    we  get   x  -  a  sm20,    — -  =  tan  0. 
2gc~  dx 

Aeain   —  =  —  —  =  2a  sin'-0  ;  hence  we  obtain,  as  the  equations  of  the 
°         dO       dx   dd 
curve,     x  =  a  sin2tf,     y  =  a  (0  —  sin  6  cos  0)  +  b,     where  a  and  b  are  arbitrary 
constants. 

The  curve  is  therefore  a  cycloid  {Differential  Calculus,  Art.  272). 

This  problem  is  one  of  great  interest  in  the  history  of  Mathematics,  as  its 
proposal  by  John  Bernoulli  in  1696  led  to  the  invention  of  the  Calculus  of 
Variations. 


Examples.  435 

2.  Prove  that  for  any  system  of  coplanar  forces  the  curve  of  quickest  descent 
is  such  that  at  each  point  the  pressure  on  the  curve  due  to  the  forces  is  equal  to 
that  due  to  the  motion. 


Here  n  =  j ,.   dx  ;  hence,  putting  5n  =  0, 


have,  after  integrating  hy  parts, 


dx\v\/l+p2J  v2  dy 

T£  «  Av  L-      i  d    /sinfl\  1         dv 

It  we  put  p  =  tan  0,  this  equation  becomes  —  I ] 

1   dsind      sin  0  (dv  dv\ 

that  is, —  (  —  +  tan  0  — 

v       dx  v-      \dx  dy  J 


cos  9   dy 
^sin0      sin  0  (dv  dv\  1         dv 


—  =0, 


COS  0    f/y 

tf0       1  (dv    .  dv  \ 

whence  cos  0  —  =  -  (  — -  sin  0  —  —  cos  0    . 

vr  a      dx        ■    n      (hJ  wu      x  ^  (dv    dy      dv  dx\ 

jy  ow   cos  0  =  — ,      sin  0  =  — ,     and  therefore   v2  —  =  v  (  —  — —  1  ; 

ds  ds  ds  \dx  ds       dy  ds  J 

dd 
also  — -  =  p,   where  p  is  the  radius  of  curvature,  and  mv~  =  2  $(Xdx  +Ydy)  ; 

hence,  substituting,  we  obtain 

mv~  dy  dx 

—  =  X l    — 

p  ds  ds 

which  proves  the  theorem  in  question. 

The  curve  of  quickest  descent  is  called  the  Brachystochrone.  The  propo- 
sition here  proved  is  a  case  of  a  more  general  theorem  in  the  Calculus  of  Varia- 
tions, for  the  discussion  of  which  the  reader  is  referred  to  Jellett's  Calculus 
of  Variations,  p.  140,  or  to  the  Encyclopaedia  Britannica,  vol.  24,  p.  86. 

3.  Deduce  Lagrange's  equations  of  motion  in  generalized  coordinates  and 
the  corresponding  equations  for  impulses  from  D' Alembert's  Principle  by  means 
of  the  Calculus  of  Variations. 

If  x,  y,  z  be  the  coordinates  of  any  particle  m,  T  is  given  by  the  equation 
T  =  2/«  {x2  4-  y-  4-  z1)  ;  but  T  can  also  be  expressed  as  a  function  of  the  gene- 
ralized coordinates  |i,  &c.,  and  velocities  £i,  &c.  As  these  two  expressions  for 
T  are  always  identical,  so  also  are  the  expressions  for  jdTdt  derived  from  them  ; 
we  have  therefore 

f  /    dZx       .  dZy      .  ddz\    ,        f  (dT  dT  dlh       „     \ 

f 5">  (• Tt  +  »* '  +  -"  * )  dt  =  j  (se  5fl  +  W>  ^  +  &c')  "'■ 

If  we  integrate  by  parts  each  side  of  this  equation,  the  terms  remaining 
under  the  integral  sign  on  one  side  must  be  equal  to  those  remaining  under  that 
sign  on  the  other,  and  a  similar  equality  must  hold  good  for  the  terms  outside 
the  integral  sign  at  each  limit.     Hence  we  have 

Id  dT     dT\  Id  dT      dT\ 

dT'  dT' 

and  -t-t  S£i'  4-  -rs-r 5£2'  4-  &c.  =  2w  (%'  5x'  4-  y  5y'  4  z'5z'). 

«|i  a|2 

2  F  2 


436  The  General  Equations  of  Dynamics. 

Since  the  limits  are  arbitrary  the  latter  equation  may  be  written 

dT  dT 

7T  5£i  +  -JT  5£a  +  &c.  =  %m  (x5x  +  y$y  +  zSz). 

«£i  d\i 

If  we  now  employ  D'Alembert's  Principle,  the  equations  of  motion  are 
immediately  obtained. 

304.  Iieast  Action. — The  integral  J"  2  Tdt  taken  between 
two  given  configurations  of  a  system  is  termed  the  Action  of 
the  system  in  passing  from  one  of  these  configurations  to  the 
other.     If  we  denote  the  action  by  A,  we  have  the  equation 


A  =  2 


Tdt,  (40) 


where  f  and  t"  correspond  to  the  initial  and  final  configura- 
tions of  the  system. 

If  v  be  the  velocity,  m  the  mass,  and  s  the  path  of  any 
particle  of  the  system,  it  is  plain  that  A  may  be  expressed 
also  by  the  equation 


A  =  S/rc      vds  =  Sm 


(xdx  +  ydy  +  zdz),       (41) 


where  s  and  s"  are  in  any  individual  motion  the  values  of  s 
for  the  particle  m  in  the  initial  and  final  configurations. 

The  Principle  of  Least  Action  asserts,  that  subject  to  the 
condition  imposed  by  the  equation  of  energy  the  mode  in 
which  a  conservative  system  passes  from  one  configuration  to 
another  is  such  that  the  action  is  a  minimum. 

The  equation  of  energy  is  T  +  V  =  E,  where  E  is  con- 
stant, and  V  a  given  function  of  the  coordinates.  This 
equation  determines  T  as  a  function  of  the  coordinates,  but 
not  v  the  velocity  of  an  individual  particle.     Hence  the  value 

fs" 
of       vds  depends  not  only  on  the  initial  and  final  positions  of 

the  particle,  but  also  on  the  relation  which  in  any  individual 
actual  motion  exists  between  v  and  s.  If  we  consider  the  ex- 
pression for  A  given  by  (40)  it  is  plain  that  the  value  of  A 
depends  on  the  equations  which  are  supposed  to  determine 
the  coordinates  in  terms  of  t  in  any  individual  motion  of  the 
system,  and  the  Principle  of  Least  Action  asserts  that  in 
the  actual  motion  of  the  system  these  equations  are  such  as 
to  render  A  a  minimum.     The  student  should  observe  that 


Least  Action.  437 

in  (40)  the  limiting  values  of  t  are  not  given.  In  fact,  when 
the  initial  and  final  configurations  are  given  the  correspond- 
ing values  of  t  depend  upon  the  actual  motion  of  the  system. 

To  show  that  A  is  a  minimum  in  the  actual  motion  we 
must  suppose  the  forms  of  the  functions  by  which  cc,  &c,  are 
expressed  in  terms  of  t  to  vary,  and  prove  that  the  consequent 
variation  of  A  is  zero. 

We  have  then  by  (38) 

DA  =  2T"dt"  -  2T'dt'  +  j2STdt. 
Now  ST  +  $V=  0,  and  therefore  we  get 

DA  =  2T"dt"  -  2T'dt'  +  J  (ST  -  dV)  dt; 
also,  since  2  T  =  'Em  (dr  +  if  +  s2) , 

we  have  $r=  2m  (i£i  +  y§y  +  z$z), 


hence 


STdt 


.  .d$x      .  dh/      .  dSi . 


-  +  my  J  Sy  +  f^?+  miA  &s  J  «.       (43) 


If  we  integrate  each  term  by  parts,  and  substitute  in  the 
expression  for  DA,  we  obtain 

DA  =  2T"dt"-2T'dt' 
+  2w(i"&*>"  +  jTW+  *"&")  -  2f»@W  +  y  V  +  s'&0 

Now  by  D'Alembert's  equation  the  part  under  the  inte- 
gral sign  must  be  zero,  and  therefore  if  the  part  outside 
that  sign  be  likewise  zero,  we  have  DA  =  0. 

But        2T'dt"  +  2/rc {x"  Ix"  +  y'ltf  +  z" Ss") 

=  2m{x"  (x'dt"  +  &*>')  +  y"  (fdt"  +  Bf)  +  z"\z"df  +  $z")  J , 

and  df'dt"  +  dx",  &c.  are  by  (39)  the  complete  variations  of 
%",  &c,  and  therefore  must  each  be  zero,  since  x'\  y\  s",  &c. 
are  invariable,  being  the  coordinates  of  the  particles  of  the 
system  in  its  final  configuration,  which  is  given.  Hence,  as 
similar  results  hold  good  for  the  other  limit,  we  obtain 
DA  =  0,  and  therefore  may  conclude  that  A  is  a  minimum 
or  a  maximum. 


438 


The  General  Equations  of  Dynamics. 


If  the  potential  energy  of  a  system  be  given  as  a  function 
of  the  generalized  coordinates,  the  Principle  of  Least  Action 
enables  us  to  arrive  at  its  equations  of  motion. 

To  obtain  the  equations  of  motion  in  this  manner  we  must 
seek  to  determine  the  generalized  coordinates  as  functions  of 
t  in  such  a  way  as  to  make  A  a,  minimum,  subject  to  the 
condition  that  T  +  V  =  constant.  This  condition  gives 
ST  +  SV=  0,  and  therefore  if  A  be  an  indeterminate  quantity 
we  must  have,  when  A  is  a  minimum, 

DA+j\($T+SV)dt  =  0.  (44) 

In  this  equation  the  variations  £|i,  &c.  may  be  regarded 
as  independent  and  arbitrary,  provided  we  can  determine  A 
so  as  to  satisfy  the  equation  T  +  V  =  constant. 

If  we  substitute  ^  ^  +  —  S&  +   &c.   for  ST  and 
d£x     dt       d|i 

-rs-  Sgi  +  &c.  for  SV  in  (44),  we  get,  after  integrating  by 

parts,  for  the  terms  under  the  sign  of  integration, 


,Q     Vr/T     .dV      d 
(2  +  A }  —  +  A 

d&       d&     dt 


(2  +  A) 


(IT 
dt 


S&  +  &o. 


<#. 


Hence,  as  the  part  under  the  integral  sign  must  vanish 
independently  of  the  terms  outside  that  sign,  and  as  <5£i,  &o. 
are  independent  and  arbitrary,  we  have  the  system  of  equa- 
tions 


/0         (dT     d  dT\        dV 

(2  +  A)    — ■ +  A  — 

\rf£i      dt  dlJ        dKi      4i  dt 


dT  d\  _     ~\ 


(2  +  A)(^ 


1  !*? 

&c. 


dV     (IT  dX 


+  A— -^-— =0   | 
d%2      dt  dt 


> 


(45) 


If  we  multiply  the  first  of  these  equations  by  fi,  the 
second  by  £>,  &c.  and  add,  we  have 


Wit 


d_dT\t 
dt  4  / 


AS 


IT 


(46) 


Hamilton's  Characteristic  Function.  439 

Hence,  by  (25)  and  (13),  we  obtain 

,<>  +  KdT+\dV    9T(/X-0 

-[2  +  1)Tt+Xlf'2Tdt-°' 
thatls'  rfF  -  2— V  It  "  27a  It  =  °-  <4'> 

This  equation  becomes  the  same  as  the  equation  of  con- 
dition T  +  V  -  constant,  provided  A  =  -  (2  +  A),  or  A  =  -  1. 
Equations  (45)  then  become  the  same  as  Lagrange's  Equa- 
tions (22).  It  is  easy  to  see  that  if  A  =- 1,  the  terms  outside 
the  sign  of  integration  in  (44),  after  integrating  by  parts, 
vanish  of  themselves  when  the  limiting  values  of  £u  £2,  &c 
are  given. 

Some  eminent  mathematicians  have  deduced  the  equa- 
tions of  motion  from  the  Principle  of  Least  Action  in  a 
strangely  illogical  manner. 

305.  Hamilton's  Characteristic  Function.  —  The 
motion  of  a  given  system  having  n  degrees  of  freedom  whose 
potential  energy  is  a  given  function  of  the  coordinates  is 
completely  determined  if  the  initial  values  of  the  generalized 
coordinates  and  velocities  be  given.  At  any  subsequent  un- 
determined time  t  we  have  n  equations  connecting  t  with  the 
corresponding  values  of  the  coordinates  and  the  2n  quantities 
previously  assigned.  If  t  be  eliminated  from  these  equations 
n  —  1  remain.  Again,  the  kinetic  and  potential  energies  are 
at  any  time  connected  by  the  equation  T  +  V  =  E,  which 
gives  another  relation  between  the  2n  assigned  quantities. 
Hence  we  conclude,  that  if  the  initial  values  of  the  coordi- 
nates be  given,  and  also  their  values  at  any  subsequent 
undetermined  time,  along  with  the  total  energy  E  of  the 
system,  the  motion  is  completely  determined. 

It  follows  from  what  has  been  said  that  the  action  A  of  a 
system  in  passing  from  one  configuration  to  another  is  a 
determinate  function  of  the  initial  and  final  values  of  the 
coordinates  and  of  the  total  energy.  This  function  is  called 
by  Hamilton  the  Characteristic  Function.  Whenever  it  can 
be  assigned  it  furnishes  us  with  the  first  and  second  integrals 
of  the  equations  of  motion,  as  we  proceed  to  show. 


440  The  General  Equations  of  Dynamics. 

Suppose  each  of  the  initial  and  final  coordinates,  as  well 
as  the  total  energy  of  the  system,  to  be  slightly  altered,  then 
each  coordinate,  at  any  intermediate  time,  receives  a  corre- 
sponding variation,  and  so  likewise  does  T,  the  kinetic  energy 
of  the  system.  Now  A  =  2  J  Tdt,  and  therefore  8  A  =  J" 28TM  ; 
but  BT+SV=  Whence 

$A=j(ST+$E-$V)dt.  (48) 

If  in  this  we  substitute  for  jSTdt  its  value  given  by  (42) 
and  integrate  by  parts,  we  find,  as  in  (43),  that  the  part 
under  the  sign  of  integration  must,  in  virtue  of  D'Alembert's 
equation,  be  zero.  Hence  $A  must  consist  entirely  of  the 
terms  outside  the  sign  of  integration.  To  ascertain  what 
these  are  when  T  is  expressed  as  a  function  of  the  generalized 
velocities  and  coordinates,  we  must  put  for  ST  in  (48)  the 
expression 

^IT  dTd3 

\d%  d%     dt 

Since  SA  as  shown  above  consists  entirely  of  the  terms 
outside  the  sign  of  integration,  if  %l9  g2,  &c,  ?/,  £2',  &c,  be 
the  final  and  initial  coordinates,  we  obtain  thus 

8A=(t-t')$E+~  S&  +  ^8g,+&0._f!*£sEi/+  ^8&'+&o.\ 

dt         d^  U/  dli  J 

dT  • 
Now  DA  =  2Tdt  -  2T'dt'  +  BA,  and  2T  =  S  -4  ?, 

d% 

hence  by  (39)  we  get 

DA  =  {t-  if)  $E  +  pMi  +P-M,  +  &o.  -  (pi'DE/  +P2D&  +  &o.) 

where  ^1,  &c.  have  the  same  meaning  as  in  (10). 

Again,  A  being  supposed  to  be  expressed  as  a  function  of 
the  initial  and  final  coordinates  and  total  energy  of  the 
system,  we  have 

_.         dA     y    dA     y     B      dA     y ,  dA     r ,     p        dA  ~  -, 
DA=  — Dgi+— DL+&0.+  —  D?i  +  —  i>s2  +&c.+  —  S-#. 


arbitrary,  we  get 

dA 

dA 

dA 

d& 

«-*•■ 

'   dln~ 

dA 

dA 

dA 

wr~Pi> 

dE~t     t' 

Hamilton' 's  Characteristic  Function.  441 

Comparing  the  two  expressions  fori) A,  and  remembering 
that  Z)£n  Dt,2,  &o.  D?/,  Df/,  &c.  and  BE  are  independent  and 


ft;  (49) 

--*.';     (50) 

(51) 

Equations  (49)  and  (51),  if  E  be  eliminated,  furnish  ex- 
pressions for  |i,  £2,  &c,  in  terms  of  the  coordinates  and  the 
time,  in  other  words,  the  first  integrals  of  the  equations  of 
motion.  Equations  (50)  and  (51),  if  E  be  eliminated,  enable 
us  to  express  the  coordinates  themselves  as  functions  of  the 
time,  and  so  furnish  the  second  integrals  of  the  equations  of 
motion.  In  each  case  the  initial  coordinates  £/,  &c,  and 
components  of  momentum  pi,  &c,  are  supposed  to  be  given. 
It  is  to  be  observed  that  if  we  desire  to  have  the  first  inte- 
grals in  their  usual  form,  in  which  the  arbitrary  constants  are 
determined  from  the  initial  velocities,  we  must  employ  all  the 
equations  (49),  (50),  and  (51),  and  eliminate  £i,  &c,  as  well 
as  E. 

In  the  case  of  a  set  of  free  particles,  equations  (49)  and 
(50)  become 
dA         .      dA         .      dA  .      dA         .     Q  /tox 

*  =  '"'*"  fe  =  myi>  25  -  "hZl>  dT2  =  mH'  &0- ;      (52) 

dA  .,  dA  .,  dA  .,  dA  . 

J2 = -"*  -  ^  =  -»* .  s> = -«•* .  sj  =  -  «* . &0-  («») 

The  function  ^1  satisfies  certain  partial  differential  equa- 
tions by  which  it  may  sometimes  be  determined.  These 
equations  are  obtained  thus  : — Multiply  the  first  of  equations 
(49)  by  &,  the  second  by  £»,  &c,  and  add,  and  we  have 

J  A   .       rl  A    . 

~  s. + 4-  & + &o- = 2T= 2  c*  -  n-     (s*) 

«S1  "S2 


442  The  General  Equations  of  Dynamics. 

In  like  manner,  from  (50)  we  get 

rl  A    •  rl  A     • 

^  ^ + «£  &,  +  &c#  =  _  2r  =  2  ( r  -  ^).     (55) 

o%(  dl% 

In  equation  (54)  we  must  remember  that  £i,  ?2,  &c.  are 
supposed  to  be  expressed  as  functions  otpi9pt9  &c,  and  thus, 
finally,  as  functions  of 

clA      clA 

~o%?     d& 

A  similar  remark  holds  good  for  (55). 

In  the  case  of  free  unconnected  particles,  equations  (54) 
and  (55)  take  the  simple  forms, 


2- 

m 


MQHfMS)]-^-' '•  m 


4PH£M»)'}-3<^  157> 


Examples. 

1 .  Find  the  characteristic  function,  and  the  initial  and  final  integrals  in  the 
case  of  a  hody  falling  vertically. 

Here  there  is  only  one  coordinate,  z  the  height  of  the  hody  from  the 
ground.  Since  gravity  tends  to  diminish  z,  the  potential  energy  V  =  mgz> 
and  E  =  T  +  mgz.     We  have,  then, 

=(£)■-•<■--  >•  =(»)->«»-^ 

where  z  is  the  initial  height.  If  we  attribute  the  negative  sign  to  the  square 
root  in  the  first  of  these  equations,  we  get,  by  integrating, 

2,cj  \         m         j 

In  this  equation  C  is  a  function  of  z,  and  is  to  be  determined  from  the  second 
differential  equation  for  A.  Remembering  that  A  must  vanish  when  z  =  z\ 
we  get  finally 


Examples.  443 

We  have,  then, 


dA  \2{E-mgz)  .,        ,        dA  J2(E 

ms=Pl  =  -d7  =  -m  V— iT- '    ""  =Pl  =  -  H  =  ~  m  <~ 


mgz') 


dA      I  (  (2(E-  mgz) \  \  _   /2(E-mgz') \  § | 
~d~E~g\  \         m         /     ~  \  w  /     i  ' 

If  we  eliminate  E  and  z  from  these  three  equations,  and  put  z"  =  -  v',  we 

get  the  ordinary  first  integral  of  the  equation  of  motion  in  which  the  initial 

velocity  is  supposed  to  he  given.     If  we  merely  eliminate  E  between  the  last 

two  of   the   above   equations,   and  put    z'  =  —  v' ,  we  get  the  ordinary  final 

integral. 

t~ 
The  resulting  equations  are     £  =  —  {gt  +  v).     z  —  —  g  —  -  v't  +  z'. 

The  signs  which  we  have  attributed  to  the  square  roots  correspond  to  the 
motion  of  a  falling  body  projected  vertically  downwards.  The  results  which 
hold  good  in  the  other  cases  of  the  motion  of  a  body  falling  vertically  are 
deduced  from  the  general  equations  by  giving  the  proper  signs  to  the  square 
roots. 

2.  A  material  particle  is  acted  on  by  an  attractive  force  passing  through  a 
fixed  point,  and  varying  directly  as  the  distance  ;  find  the  characteristic 
function. 

Let  m  be  the  mass  of  the  particle,  and  fir  the  magnitude  of  the  force  at  the 
distance  r,  then 

dV  3        rr       A*,    o 

~~  ~dx   =  ~  ftX'  2  *      +  r ) ' 

Hence  we  have 

(a,+(S),-tM,-^+^>-    w 

If  we  assume 

the  equation  (a)  is  satisfied,  provided 

ci  +  c2  =  IE.  (e) 

Since  the  differential  equation  to  be  satisfied  bv  77  and  -—  is  similar  to  (a), 

'  dx  dij 

and  since  A  must  vanish  when  x  =  x'  and  y  =  y',  we  have 


444  The  General  Equations  of  Dynamics. 

In  this  expression  for  A  the  constants  cx  and  c%  are  subject  to  the  condition 
ci  +  c%  =  2E.  In  order  that  A  should  he  expressed  as  a  function  of  x,  y,  x\  y\ 
and  JE,  a  second  equation  connecting  c\  and  c*  with  these  quantities  is  required. 
This  equation  is,  in  fact, 


x  A—  —  sin-1  x'l—=  sin-1  y  J sin-1  y'J- 

\  <?1  \  Ci  ^  Co  *  Co 


(•) 


Its  truth  may  he  proved  as  follows  : — 

By  equation  (d)  A  is  expressed  as  a  function  of  x,  y,  x',  y',  ci,  Co,  so  that 
we  may  write  A  =  <p  (x,  y,  x',  y',  a,  Co).  An  equation  must  exist  between 
<-\,  C2,  x,  y,  xf,  y\  by  means  of  which  <p  can  be  transformed  into  \p(x,  y,  x',  y',  a  +  c2). 
We  have,  then, 

d\p       d<p       d<p  dco       d\p      d(p      d(p  dc\  dty  _  cty 

dc\      dc\      dc2  dc\      dc-i      dc-2      dc\  dc%  dc\      dc% 

and  therefore  ft  ( 1-p)  =  ft  ( 1  -ft),  that  is,  ft  *,+  ft  **  =  0. 

d<p       dd> 
Again,  dc\  +  dc%  -  0,  since  c\  +C2  =  2.E',  and  therefore  we  have  —  =  — . 

aci      aC2 

Hence  the  required  relation  between  c\  and  c2  must,  in  virtue  of  (c),  be  capable 

dd>      deb      _,  „      ffo        ,  ^</> 

of  being  expressed  in  the   form,  -f-  =  -z.     The  expressions  for  —   and  — - 

are  found  most  easily  from  (b).     From  these  equations  we  have 

dA         /-     / c?L4  a/w 

—  =  Y  m  \/  ci  —  /j.  x£,     whence 


Integrating,  we  have 

dA  ,-  [x  dx  \m  /  U        .    .    ,    Ifi  \ 

In  like  manner 

dA  \m  I  .    ,       L        .    ,    ,     lju.\  . 

—  =7A  -    sin  m/.  -  —  sin  l  y  \  - )  > 
dco      2\^  V  >^2  y    >c2/ 


,  .  <L4      dA 

hence,  since    — ■  =  -—.  we  have  (e). 
dci      dc2 


(     445     ) 


CHAPTER  XIII. 


SMALL     OSCILLATIONS. 


306.  Introductory  Considerations. — When  a  material 
system  in  equilibrium  under  the  action  of  any  forces  is 
slightly  disturbed,  the  several  points  of  the  system  in  many 
cases  tend  to  return  to  their  original  positions.  In  such 
cases,  if  the  distance  of  each  point  from  its  position  of  equi- 
librium remains  during  the  motion  very  small  as  compared 
with  the  other  magnitudes  on  which  the  motion  depends, 
the  system  performs  small  oscillations. 

Some  cases  of  small  oscillations  have  been  already  con- 
sidered in  Articles  102  and  193.  The  simplification  of 
the  problem  in  the  case  of  small  oscillations  has  been  exem- 
plified in  the  articles  referred  to,  and  consists  in  neglecting 
the  squares  and  higher  powers  of  small  quantities. 

Before  proceeding  to  the  general  theory  of  small  oscilla- 
tions we  shall  illustrate  the  method  by  the  consideration  of  a 
few  elementary  cases. 

307.  Oscillation  on  a  Plane  Curve. — We  commence 
with  the  small  oscillation  of  a  particle,  under  the  action  of 
gravity,  on  a  smooth  vertical  circle. 

Taking  the  lowest  point  on  the  circle  as  origin,  the 
vertical  diameter  as  axis  of  s,  and  the  tangent  as  that  of  x9 
the  equation  of  the  circle  is 


2az 

=  x*  +  z\ 

(i) 

where  a 

is  its  radius. 

Also, 

by  D'Alembert's 

principle 

(Art. 

196) 

we 

have 

xh>  +  «&  +  gh  =  0.  (2) 

Now,  for  a  small  oscillation  x  must  be  small  throughout 
the  motion,  and  consequently  s  is  a  small  quantity  of  the 
second  order. 


446  Small  Oscillations. 

Hence,  to  the  degree  of  approximation  required,  we  have 
adz  =  x$x,     and.  az  =  xx ;     therefore  az  =  xx  +  or  ; 
we  may  accordingly  neglect  z,  and  equation  (2)  becomes 

x  +  g-  x\  Ix  =  0,    or  x  +  9-  x  =  0. 

The  integral  of  this  equation  is 

*  =  *sin(^  +  X),  (3) 

as  in  Art.  102. 

In  like  manner,  if  any  curve  be  taken  instead  of  the  circle, 
its  equation,  referred  to  the  tangent  and  normal  at  its  lowest 
point,  may  be  written 

2s  =  c0%2  +  2cYxz  +  2c2z2  +  &c. 

Accordingly,  neglecting  terms  of  a  higher  order  than  the 
second,  we  have  gs  =  c&$x,  and  it  is  readily  seen  that  z  may 

1 
be  neglected  as  before  ;  also  observing  that  e0  =  -  {Biff.  CaL, 

Art.  230),  where  p  is  the  radius  of  curvature  at  the  origin, 
we  get  immediately  from  (2), 


/*•  sin  f  t 


■IH- 


This  shows  that  in  all  such  cases  the  motion  is  represented 
by  a  simple  harmonic  function. 

308.  Oscillation  on  a  Smooth  Surface. — We  shall 
next  consider  the  case  of  a  small  oscillation,  under  gravity, 
on  a  smooth  spherical  surface. 

Taking  the  origin  at  the  lowest  point  on  the  sphere,  and 
the  z  axis  vertical,  the  equation  of  the  sphere  is 

2az  =  x2  +  f  +  z\  (4) 

Also,  from  D'Alembert's  principle, 

x§x  +  i/hj  +  z$z  +  g§z  =  0.  (5) 


Oscillation  on  a  Smooth  Surface.  447 

Here  we  may  neglect  z2  and  z  as  before,  and  thus  we  obtain 
immediately 

(*+f.)&  +  (#+j[y)*-o. 

Hence  x  +  —  x=  0,     j)  +  -  y  =  0  ; 

c i  ci 

accordingly  we  have 

x  =  m  sin  [t^g-  +  x\      y  =  n  sin  [t^  +  x. 

where  m,  n,  ^1?  X2  are  arbitrary  constants. 

These  equations  may  also  be  written  in  the  form 


x  =  a  sin  t  I-  +  a  cos  t  /- 
\a  \a 

,j  =  j5  sin  <J?+  ,3'  cos  tjl 


(6) 


in  which  a,  /3,  a',  j3r  are  small  arbitrary  constants,  whose 
values  depend  on  the  initial  circumstances  of  the  motion. 

Hence,  if  the  particle  be  set  in  motion  with  a  small 
initial  velocity  from  a  point  near  the  lowest  point  on  a  sphere, 
its  motion  will  be  given  by  equations  (6). 

Also,  if  we  eliminate  t  from  these  equations,  we  see  that 
the  horizontal  projection  of  the  path  of  the  particle  is  an 
ellipse.     (Compare  Art.  193.) 

We  shall  now  consider  the  oscillatory  motion  of  a  particle, 
under  gravity,  on  any  smooth  concave  surface. 

Neglecting,  as  in  the  former  cases,  small  quantities  of  a 
higher  order  than  the  second,  the  equation  of  the  surface, 
when  referred  to  the  normal  and  tangent  plane  at  its  lowest 
point,  may  be  written 

2s  =  ax2  +  2hxy  +  hf.  (7) 

This  gives         Zz  =  (ax  +  hy)  Sa?  +  (hx  +  by)Sy. 


448  Small  Oscillations. 

Also  z  may  be  neglected,  as  before,  and  equation  (5)  becomes 

{x  +  g  (ax  +  hy) }  Ix  +  \y  +  g  (fix  +  by) }  $y  =  0. 

Hence 

x  +  g  (ax  +  hy)  =  0,      y  +  g  (hx  +  by)  =  0.  (8) 

Now,  these  being  linear  differential  equations,  we  may  put 

x  =  m  sin  (t  </\  +  x),    V  =  w  sin  (£^/A  +  x) ; 
this  leads  to  the  equations 

(ga  -  A)  m  +  ghn  =  0,     grto  +  (gb  -  A)  w  =  0. 
Accordingly  A  must  be  a  root  of  the  equation 
ga  -  A,     gh 
gh,      gb-\ 


0,  (9) 


A  -  ga 
and  n  =  —  m. 

Hence,  if  Ai  and  A2  be  the  roots  of  (9),  we  see  that  the  com- 
plete integrals  of  (8)  may  be  written 

x  =  mi  sin  (t^/Xi  +  xO  +  m* sm  if  \/^  +  x»)  ) 

\-   a  -  j(1°) 

=AL_|«  ^  gin  ^  y Ai  +      )  +  _2^«  mj  gin  ^  yA2  +  Xoj 

J        gh  g/l  ' 

in  which  mi,  m2,  xu  X2  are  arbitrary  constants,  of  which  the 
two  former  must  be  very  small,  in  order  that  the  motion 
should  be  one  of  small  oscillation.  It  is  readily  seen  that 
this  solution  would  fail  if  either  Ai  or  A2  were  negative  ;  thus 
if  A  be  negative,  instead  of 

wsj  sin  (t  v^Ai  +  x0> 

we  shall  have  the  terms 

where  jui  =  -  Ai» 


Oscillation  on  a  Smooth  Surface.  449 

The  motion  will  then  not  be  a  small  oscillation,  as  this 
expression  will  increase  continually  with  t,  unless  in  the 
exceptional  case  where  hi  =  0. 

Again,  if  Rl  and  R2  be  the  principal  radii  of  curvature  at 
0,  it  is  readily  seen  that 

A   -  9       \  -  g 

For  let  the  ellipse  ax2  +  2hxy  +  by2  =  c  be  transformed  to  its 
axes,  so  that 

ax2  +  2hxy  +  bif  =  a'X2  +  b'Y2 ; 

then,  since  a  +  b  =  d  +  b',     and     ab  -  h2  =  a'b', 

the  equation         (ga  -  \){gb  -  A)  -  cfh2  =  0 

becomes    -  {ga  -  A)  {gbf  -  A)  =  0. 

The  roots  of  this  equation  are  gaf  and  gtt ;  but,  as  in  Art.  307, 
we  have 

«-  =  !     6'=i 
R\  H2 

accordingly  for  a  small  oscillation  both  Rx  and  R2  must 
be  positive,  i.e.  the  surface  must  be  convex  towards  the  plane 
of  xy.  If  Ai  =  A2,  we  have  Ri  =  R2i  and  the  origin  is  a  point 
of  spherical  curvature.  In  this  case  a  small  oscillation  is 
the  same  as  on  the  surface  of  a  sphere,  and  is  given  by 
equations  (6). 

Examples. 

1 .  A  bar  of  mass  m  hanging  freely  from  one  extremity  is  slightly  displaced ; 
determine  its  motion. 

Take  two  horizontal  lines  at  right  angles  to  each  other  passing  through  the 
point  of  suspension  for  axes  of  x  and  y.  Let  the  small  angular  displacements  of 
the  bar  at  any  time  round  each  of  these  axes  towards  the  other  be  6  and  <p ; 
then,  r  being  the  distance  of  any  point  of  the  bar  from  its  extremity, 


therefore 


X  = 

r<p, 

V  =  r9, 

z*  =  r°~ 

-  x~  — 

y*; 

z  =  r{\ 

-  w 

-W) 

,     5a  «- 
2  G 

r(6S9 

+  4>80), 

450  Small  Oscillations. 

thus  we  may  neglect  z,  and  have 

Sdmr-  {6  56  +  <p8(p)  +  gldmr  {6  $6  +  <p  Sp)  =  0 . 
Hence,  i£  jr2dm  =  mk2,  jrdm  =  ml,  we  have 

I  I 

6  +  y-d  =  0,     <t>  +  ff-(p  =  0, 


(■[v'V  +  Xji     <|)  =  ^sin^v/^+x2J. 


2.  Two  balls  connected  by  a  horizontal  bar,  whose  mass  may  be  neglected, 
are  suspended  by  two  vertical  cords  of  equal  length.  The  bar  receives  a  slight 
displacement  of  rotation  round  a  vertical  axis  midway  between  the  cords  ;  find 
the  motion  of  the  system. 

Let  if/  be  the  angle  which  the  bar  makes  with  a  horizontal  line  parallel  to  its 
initial  position,  6  the  inclination  of  one  of  the  cords  to  the  vertical  (see  figure  in 
Ex.  4,  Art.  244),  I  its  length,  b  the  distance  from  the  middle  point  of  the  bar  to 
one  of  the  balls  ;  then 

x  =  bcosip  =  b{l-lf~),    y  =  bi/,     z  =  I (1  -  \62) ; 

52    „\ 


but  »  =  ty;     .-.z  =  l(l-±-rpA, 

and  x'  =  -  x,     y'  =  -y 


then  x,  x,  z,  z'  may  be  neglected,  and  equating  to  cipher  the  coefficient  of  5\p  in 
D'Alembert's  equation,  we  have 


£  +  |*  =  0; 


therefore 


{&+*) 


where  a  and  %  are  arbitrary  constants. 

This  shows  that  the  period  of  vibration  is  the  same  as  that  of  the  pendulum 
whose  length  is  I. 

3.  A  heavy  bar  is  suspended  and  displaced  as  in  the  preceding  example  ; 
investigate  its  motion. 

Let  r  be  the  distance  of  any  point  of  the  bar  from  its  centre,  and  b  the 
distance  from  its  centre  to  the  point  of  attachment  of  one  of  the  cords ;  then, 
as  in  the  preceding  example, 

J8 

\p  JY2  dm  +  g  —  i//  jdm  =  0  ; 


therefore  \f  =  a  sin  (  7  -Jt  *  +  X  )>  where  J>2  dm 


mk- 


Stable  Equilibrium.  451 

4.  How  must  the  bar  in  the  preceding  example  be  suspended  in  order  that 
its  vibrations  should  be  isochronous  with  those  of  a  ball  hung  by  one  of  the 
supporting  cords  ? 

Ans.    b  =  k.     In   the   case   of  a  homogeneous   bar    whose   length  is  2a 

V3 

5.  A  uniform  rod  of  mass  m  hangs  from  a  horizontal  pivot  passing  through 
one  of  its  extremities.  An  inextensible  string,  whose  weight  is  negligible, 
attached  to  the  other  extremity,  passes  through  a  smooth  ring  situated  on  the 
vertical  line  through  the  pivot  at  a  distance  below  it  equal  to  the  length  of  the 
rod,  and  sustains  a  mass  p.  The  rod  being  slightly  displaced  from  its  position 
of  equilibrium,  determine  the  motion. 

The  equations  of  motion  are 

%  ma2  9  =  -  mga  sin  9  —  2a  T,    p  z  =  T  —  pg, 

where  a  is  half  the  length  of  the  rod,  and  z  the  vertical  coordinate  of  p.  If  z 
be  measured  from  the  position  of  p  when  the  rod  is  vertical,  z  =  4a  sin  ±-9. 
Since  9  is  always  small,  we  may  take  sin  9  =  9  ;  substituting  for  z  and  elimi- 
nating T,  we  have 

|  a{m  +  3p)  9  +  nig  (9  +  —  \  =  0. 
Hence  the  rod  returns  to  its  vertical  position  in  a  time 

l[4a(u3p)}coa-i     2p 

\j  1 3g  \        m  J  )  2p  +  m9o 

where  0o  is  the  initial  value  of  9. 

309.  Stable  Equilibrium. — A  position  of  stable  equi- 
librium is  one  from  which  a  system  has  no  tendency  to  depart 
far  if  it  be  slightly  disturbed. 

In  a  conservative  system  if  the  potential  energy  be  a 
minimum  the  corresponding  position  is  one  of  stable  equi- 
librium ;  as  may  be  shown  in  the  following  manner  : — 

From  equation  (4),  Art.  282,  we  have  T  +  V  -  V0  =  T0. 
Now  since  T  =  \  ^mv2  it  is  always  essentially  positive  ;  also, 
V0  being  the  minimum  potential  energy,  V  -  V0  is  positive 
for  all  small  values  of  the  variables,  and  may  therefore  be 
reduced  to  a  number  of  squares  with  positive  coefficients. 
Therefore  if  T0  be  small,  each  term  both  of  T  and  of  V  -  V0 
must  be  small,  and  must  always  remain  so.  Hence,  if  the 
original  disturbance  be  slight  the  system  can  never  depart 
far  from  the  position  of  equilibrium  nor  attain  a  high  velo- 
city.     The  position  is  therefore  one  of  stable  equilibrium. 

2  G  2 


452  Small  Oscillations. 

310.  Equations  of  Motion  for  an  Oscillating  Sys- 
tem.— In  the  following  investigation  of  the  small  oscillations 
of  a  system  ahout  its  position  of  equilibrium,  it  is  assumed  that 
the  forces  which  act  at  the  different  points  of  the  system  are 
functions  of  the  coordinates  of  those  points,  and  that  the 
constraints  and  mutual  connexions  can  be  expressed  by  means 
of  equations  between  the  coordinates. 

In  virtue  of  these  equations  the  coordinates  of  the  points 
of  the  system  are  functions  of  n  independent  variables,  and 
these  again  are  at  any  time  functions  of  their  values  in  the 
position  of  equilibrium,  and  of  the  increments  resulting  from 
the  disturbance  from  this  position  and  subsequent  motion. 
If  the  system  perform  small  oscillations  the  increments  of 
the  variables  are  all  small  quantities,  whose  squares  and 
higher  powers  may  be  neglected. 

Hence  the  equations  of  motion  involve  only  the  first 
powers  of  the  variables  and  of  their  differential  coefficients. 
In  other  words,  they  form  a  system  of  linear  differential 
equations  with  constant  coefficients. 

Let  ai,  a2,  ...  an  represent  the  values  of  the  generalized 
coordinates  in  the  position  of  equilibrium,  and 

<t\  +  ?i,    a2  +  ?2,    .  .  .  a*  +  £n 

their  values  at  any  time  during  the  motion.  Then  x,  y,  z 
being  the  Cartesian  coordinates  of  any  point  of  the  system, 
we  have 

x  =f(al  +  ?i,  a2  +  £8,  . .  .  an  +  £n) 

=/(a1?  «2,  .  .  .  an)  +£&  +  &&...+  £.&+  &c,    (11) 

aax  da2  dan 

whence,  differentiating,  we  get 

.       df  s.       df  p  df  * 

a  a  1  da%  dan  K     ' 

since  the  squares  and  higher  powers  of  %Xy  &c.  may  be 
neglected  ;  similar  equations  hold  good  for  ij  and  z.     Hence 


Equations  of  Motion  for  an  Oscillating  System.        453 

T,  the  kinetic  energy  of  the  system,  is  a  quadratic  function 
with  constant  coefficients  of  f  t,  j2,  &c.,  and  we  may  write 

2T=fnV  +M**  +  &c.  +  2/12ya  +  &c.  (13) 

Again,  if  V  be  the  potential  energy,  we  have 

V  =  i^(ai  +  Ei,   a2  +  &,...  a„  +  ?»). 

Expanding  by  Taylor's  Theorem,  putting  F0  =  JP(ai,  a2, . . .  a„), 
and  neglecting  powers  of  &,  &c.  higher  than  the  second,  we 
get 

^Fop     f/F0~  ^F0p      J^Fo.j     „    \     ,... 

F=F0+  —  &+-T—6I.  •  -  +  -7—  E„  +  |  -r-rEi2+&o.  .     14 

aa!  aa2  dan  \ciai  J 

Now  since  V0  is  the  potential  energy  of  the  system  in  a 
position  of  equilibrium,  3  V0  =  0  for  all  possible  variations  of 
ai,  a2,  •  • .  aM,  and  since  these  variations  are  independent  and 
arbitrary,  we  must  have 

dVo    n    dV0  dVo     n  ,_, 

7—  =  0,    —  =  0,  . . .  -—  =  0.  (lo) 

Hence,  if  we  put 

cPVo  <PVo  <PVo  o  na, 

■gjr-to   sj-fc   f7^2=^,&c,  (16) 

(14)  becomes 

F  =  V0  +  §  fengi2  +  &■&■  +  2^i?2  +  &c.)  (17) 

If  we  substitute  the  values  of  T  and  F  given  by  (13)  and 
(17)  in  Lagrange's  Equations  (22)  Art.  297,  we  obtain 

/11I1  +/ia|a  ■  .  •  +/u»f»  +  firii£i  +  S'wSa  •  •  •  +  2it£n  =  0\ 

fizZi  +/22?2  •  •  •  +/an|n  +  ffwEi  +  ?»&  •  •  •  +  ?2„L  =  0 


.(18) 


/u»5l"+/t»ia  •  •  •  +fnnL  +  ginSi  +  V*£*  •  •  •  +  2W?«  =  0 


454 


Small  Oscillations. 


311.  Solution  of  Equations  of  Motion — Harmonic 
Determinant. — As  in  Art.  308,  if  we  assume 

%i  =  Kax  sin  (t  a/A  +  x)»     ?« =  Ka*  s^n  ft  v  X  +  x)>  &c-> 

and  substitute  in  the  equations  of  motion,  we  obtain  the 
n  equations 

(/nX  -  ?ii)  «i  +  (/12X  -  qu)  at  . . .  +  (/mX  -  ?1M)  an  =  0  \ 
(/12X-  0ia)  0]  +  (./22X  -  g22)  0,  . . .  +  (/2MX  -  ^  an  =  0 


(/inX  -  qm)  «l  +  (/anX  -  ?2H)  flfe  .  .  .  +  (/„„X  -  ^nn)  «»  =  0 


(19) 


These  can  be  satisfied  by  the  ratios  of  the  n  determinable 
constants  «1?  a29  . . .  #»,  provided  X  is  a  root  of  the  equation 


,/nX  -  £11,    ,/i2X  -  qu9  .  . .  fmA  -  qxn 
/12X  —  ^12,    /22A  -  §,22>  •  •  •  y  2nX  —  q%n 


,/i«X  —  g'm,    ,/oMA 


y  «»X  —  gv 


=  0. 


(20) 


The  symmetrical  determinant  which  enters  into  this 
equation  we  shall  call  A.  It  is  usually  termed  the  harmonic 
determinant  of  the  motion. 

If  the  roots  of  the  equation  A  =  0  be  all  real  and  positive, 
and  be  denoted  by  A,,  A2,  .  .  .  X»,  the  complete  values  of 
Si,  £2,  &c.,  are  given  by  the  equations 


!i  =  /ci0nsin(*VAi+xi)+":2«i2sin(*VA.2  +  X2)  •  •  •  +  «««in sin  (*VaI +%»)' 
I2  =  «i«2i  sin  (t  Vai  +  xi)  +  «2«22  sin  (£  Va2  +  X2)  •  •  •  +  «n«2n  sin  (ty/  A„  +  X«) 

In  =  «1%1  Sin  (^Ai  +  x0  +  K2«;«2  sin  (*Va2  +  X2)  •  •  •  +  Knflnn  sin  (W\n+Xn)  t 


,(21) 


Lemma  in  the  Theory  of  Determinants.  455 

where  ki,  %i?  *2>  X2>  &c-  are  arbitrary  constants,  2n  in  number, 
and  On,  a2i,  .  .  .  anl  satisfy  the  n  linear  equations  for  alt  a2y . . .  an 
obtained  by  putting  Ax  for  A  in  (19) ;  al2,  a22i  .  .  .  an2  those 
obtained  by  putting  A2  for  A  ;  and  so  on. 

If  any  root  of  the  equation  A  =  0  be  real  and  negative, 

instead  of  Kian  sin  (t  */\1  +  ^0,  there  will  be  in  &  a  term  of 
the  form 

dn{icieui^  +  /ciV^'fTij, 

where  fix  =  -  \x ;  and  there  will  be  corresponding  terms  in 
£2,  £3,  &c.  In  fact  if  we  substitute  klOl^IL  for  £i>  KxOt^'v-  for 
?2j  and  so  on  in  the  equations  of  motion,  we  get  a  system  of 
equations  which  differ  from  (19)  in  having  -  fi  instead  of 
A,  and  which  can  therefore  be  satisfied  by  «i :  a21  &c.  pro- 
vided -  fi  be  a  root  of  the  equation  A  =  0.  Corresponding 
therefore  to  every  real  negative  value  of  A  there  is  a  real 
positive  value  of  /u.  In  this  case,  since  %l9  £2,  &c.  contain  in 
general  terms  increasing  without  limit  with  the  time,  the 
motion  cannot  consist  of  small  oscillations. 

If  we  suppose  au  a2,  .  .  .  an  substituted  for  &,  ?2, . . .  %n  in 
T,  and  for  %u  ?2,  •  •  •  S»  in  F,  and  denote  the  results  of 
these  substitutions  by  T'  and  V\  equations  (19)  may  be 
written 

^(Ar-n-o,^(xr-n-o,...^(Ar-n-o.  (22) 

312.   Lemma  in  the  Theory  of  Determinants. — If 

A  be  any  determinant,  and  if  the  determinants  obtained  by 
erasing  the  first  row  and  first  column  of  A,  the  second  row 
and  second  column,  the  first  row  and  second  column,  the 
second  row  and  first  column,  be  denoted  by  An,  A22,  -  A12,  -  A21, 
and  if  also  the  determinant  formed  by  erasing  the  first  row 
and  first  column  of  Au  be  denoted  by  Aim,  then  it  is  a  well- 
known  property  of  determinants  that 

An  A22  -  An  A21  =  AAiM.  (23) 

For  the  convenience  of  the  student  we  shall  give  here  a 
proof  of  this  theorem. 


456  Small  Oscillations. 

If  we  have  the  n  linear  equations 

auxx  +  a12Xi  +  ax3x:i  .  .  .  +  axnxn  =  yx 
«2i^i  +  a22x2  +  a^ors  .  .  .  +  a2nxn  -  y2 
aiXxx  +  aZ2x2  +  a^x-i  .  .  .  +  amxn  =  yz 
anxxx  +  an2x2  +  anzxz .  .  .  +  annxn  =  yn 


(24) 


and  solve  for  xx,  &c,  in  terms  of  yu  &c,  we  get  another 
system  of  n  equations,  of  which  the  first  two  are 

Axx  =  Au^i  +  A21y2  +  &c, 
Ax2  =  Ax2yx  +  A22y2  +  &c. ; 

whence,  eliminating  y2i  we  obtain  between  the  n  +  1  variables 
xn  x*9  Pn  Vzy  -  •  •  Vn  the  linear  equation 

A(A22^i  -  A2i^2)  =  (An  A23  -  A\2A2X)yx  +  &c.         (25) 

Again  we  may  obtain  a  linear  equation  between  the  same 
variables  in  another  way,  viz.,  by  eliminating  x3,  a?4, . . .  xn  from 
the  (n  -  1)  equations  got  from  equations  (24)  by  omitting  the 
second.     The  result  of  this  elimination  is 


(26) 


(27) 


axxxx  +  ax2x2  -  yx,    axz,    au,  . . .  aln 
a3Xxx  +  a32x2  -  ySf    a^,    au,  . . .  a3n 

an\X\  +  an2x2  —  yn)   #n3,    an^  . . .  ann 
which  expanded  becomes 

A22#i  -  A2V^2  =  AXX22yx  +  &c, 

Since  only  one  linear  equation  can  exist  between  n  +  1  vari- 
ables of  which  n  are  independent,  (27),  when  multiplied  by 
A,  must  be  identical  with  (25).     Hence  we  have 

AA1122  =  A11A22- A12A21. 

In  the   case  of  the  harmonic   determinant,  since  it   is 
symmetrical,  we  have  A21  =  Ai2,  and  therefore  (23)  becomes 

(28) 


Reality  of  the  Roots  of  the  Harmonic  Determinant  Equation.  457 

313.  Transformation  of  the  Harmonic  Determi- 
nant.— If  we  denote  the  quadratic  function  of  n  variables 

by  P'and  the  function 

U<h&2  +  qjtf  +  2fcA&  +  &c), 

by  <S,  the  harmonic  determinant  A  is  the  discriminant  of 
X&-  @,  and  the  equation  A  =  0  is  therefore  unaltered  by 
linear  transformation  of  the  variables  in  &  and  €>. 

Again,  when  fi,  |2,  &c,  are  substituted  for  the  variables 
in  «9[it  becomes  the  kinetic  energy  T  of  the  system.  Now, 
|i,  |2,  being  generalized  components  of  velocity,  whatever 
small  values  be  assigned  to  them,  these  values  will  belong  to 
a  possible  motion  of  the  system.  Hence  the  quantic  ^is 
positive  for  all  real  values  of  the  variables,  and  may  there- 
fore be  transformed  into  the  sum  of  n  positive  squares.  If 
this  transformation  be  effected  we  have 

2^=  m2  +  *•*  +  *•  •  •  -  +  nn\  (29) 

2©  =  sUTh2  +  522)]o2  +  2s12ijin2  +  &c.,  (30) 

and  the  harmonic  determinant  is  given  then  by  the  equation 

A—  Sn,     —  «Si2,     —  §13,      ...  —  Sm 
A=  ~  812,    X  -  S22,    -  S23,      ...  -    S%n        m  (31J 

~~  SlMj      ~"  $2»j      ~  $3»,       •  •  .  A  —  S/j^ 

314.  Reality  of  the  Roots  of  the  Harmonic  Deter- 
minant Equation. — If  the  first  row  and  first  column  of  the 
harmonic  determinant  be  erased,  and  a  similar  process  be 
applied  to  the  determinant  so  obtained,  and  again  to  the 
determinant  thus  formed  from  it,  and  so  on,  we  get  a  series 
of  determinants  beginning  with  the  harmonic  determinant 


458  Small  Oscillations, 

itself,  whose  degrees  in  X  are  n,  n-1,  n  -  2,  &c,  and  which 
in  the  present  Article  will  be  denoted  by  A»,A»_i,  .  .  .  Ai. 

It  is  to  be  observed  that  An»  An-u  An-2  are  identical 
with  A,  An,  A 1122,  and  that  Ai  is  simply  X  -  snn-  If  we  place 
+  1  at  the  end  of  this  series  of  determinants  we  obtain  a  set  of 
(n  +  1)  quantities,  such  that  when  any  one  intermediate  be- 
tween the  first  and  the  last  vanishes,  the  two  on  each  side 
of  it  take  opposite  signs.  When  A«_i  (that  is  An)  vanishes 
this  appears  from  (28),  and  it  is  plain  that  a  similar  equation 
holds  good  for  any  three  successive  determinants  in  the 
series.     Its  last  three  terms  are, 


A  —  S(»_i)  (w-i)j  —  5(n-i)» 


,    l, 


of  which  the  first  is  negative  when  X  -  snn  =  0. 

If  now  we  substitute  +  oo  for  X,  each  term  in  the  series 
is  positive,  and  if  we  substitute  -  go  the  terms  are  alter- 
nately positive  and  negative.  Hence  n  variations  of  sign  in 
the  successive  terms  of  the  series  have  been  gained  in  the  pro- 
cess of  diminishing  X  from  +  go  to  -  go  ;  but,  since  when  one 
of  the  intermediate  terms  vanishes  no  variation  is  lost  or 
gained,  a  variation  can  be  gained  only  by  passing  through 
a  root  of  the  equation  An  =  0.  In  this  way,  therefore, 
n  variations  must  have  been  gained.  Hence  the  n  roots  of 
the  equation  A«,  =  0  are  real,  and  a  variation  is  gained  in 
passing  through  each. 

From  this  last  observation  it  follows  that  when  An  first 
vanishes  A«_i  is  positive,  and  that  it  must  become  negative 
before  An  vanishes  a  second  time,  then  again  become  positive 
before  A»  vanishes  a  third  time,  and  so  on.  Hence  the  roots 
of  the  equation  A»_i  =  0  separate  those  of  A»  =  0.  In  like 
manner  the  roots  of  the  equation  A«_3  =  0  separate  those 
of  A„_i  =  0,  and  so  on. 

If  we  denote  by  t&  and  jB  the  quantics  obtained,  from 
&  and  @,  by  omitting  all  terms  containing  &  the  minor 
determinant  A«_i  belonging  to  A»  in  its  most  general  form, 
as  written  in  equation  (20),  is  the  discriminant  of  \ST-  JB, 
and  the  special  form  of  A«_i,  considered  in  this  Article,  is  the 


Stability  of  Motion.  459 

discriminant  of  the  same  quantic  after  linear  transformation. 
Hence  the  general  and  special  forms  of  A»_i  vanish  for  the 
same  values  of  A,  and  we  conclude  that  in  general  the  roots 
of  the  equation  A„_i  =  0  separate  those  of  A»  =  0.  It  is 
obvious  that  similar  considerations  apply  in  the  case  of 
A„_2,  &c 

The  results  in  this  Article  might  have  been  obtained 
directly  for  the  determinants  An,  A»_i,  &c,  in  their  most 
general  form  by  using  the  conditions  which  must  be  fulfilled 
(Biff.  Cede,  p.  460)  when  the  quantic  ^is  always  positive. 

315.  Stability  of  the  Motion. — If  we  make  A  zero  in 
the  series  of  determinants  A»,  A«_i,  &c.  of  Art.  314,  we 
obtain  a  new  series  which  may  be  denoted  by  (-  l)n  Bm 
(-  l)n_1  !)„_!,  &c.,  where  Bn  is  the  discriminant  of  @,  and  the 
remaining  determinants,  D„_u  &c.  are  formed  from  Bn  by  a 
process  similar  to  that  employed  in  obtaining  the  former 
series. 

It  is  clear,  from  Art.  314,  that  the  number  of  positive 
roots  of  the  equation  A?i  =  0  is  equal  to  the  number  of  vari- 
ations of  sign  in  the  successive  terms  of  the  series  (-  l)n  Bn, 
(-l^ZU,  .  .  .  -  A,  1. 

Hence  it  follows  that  if  Bn,  Bn.x,  &c.  be  all  positive,  the 
harmonic  determinant  equation  has  n  positive  roots.  We 
conclude,  therefore  (Biff.  Calc,  p.  460),  that  in  order  that  the 
roots  of  this  equation  should  be  all  positive,  the  quantic  @ 
must  be  positive  for  all  values  of  the  variables,  and  vice  versa. 

Without  assuming  the  truth  of  the  conditions  referred  to, 
we  may  obtain  the  same  result  in  another  way  by  employing 
the  following  transformation  :  — 

We  shall  suppose  that  &  and  @  are  of  the  form  given  by 
equations  (29)  and  (30),  and  that  the  roots  of  the  equations 
A  =  0  are  all  unequal. 

Apply  a  linear  transformation  which  will  change 

r\x  +  r)z  -f  y\,j  +  &o.    into    Z\   +  ?22  +  Ss2  +  &c., 
and  at  the  same  time  reduce  @  to  its  canonical  form 

Pxfr  +  P.JV    .    .    .    +P»?»2. 


460 


Small  Oscillations. 


In  order  to  show  that  it  is  possible  to  do  this  by  a  real  trans- 
formation, assume 


vi  =  viZ\  +  m"?2  +  »h'"?3  +  &c. 

>?»  =  J?n'?i  +  W£a  +  »7n  "?3  +  &C. 


(32) 


where  the  ratios  >?/  :  77/ :  m  :  &c.,  are  determined  by  the  equa- 
tions 


*n»h'  +  SiM*   +  s^h     •    •    •   +  smVn  ~  t\\Vi 

512*?/  +  522172'  +  S23»?/    .    •    •    +  S2nr??/  =  AilJ2 
Si«*?/  +  S2nr}2  +  Sands'    .   .    -    +  Snnr\n  =  AiTfo 

the  ratios  if/'  :  *j2"  :  17/"  :  &c  by  the  equations 

Snifi"  +  Sl2T)"   .    .    .    +  SmVn     =  A3»7i" 

_•//.-"  .    ~        ft      \      ft 

Si2*h    +  S22172     .   •   •  +  s2«r?rt    =  A2?j2 

«i»»)i"  +  s2„ij/' .  .  .  +  s„»ij»"  =  A2i?»" 


,    (33) 


k   (34) 


and  so  on. 

From  equations  (30)  and  (33)  it  follows  that  when  \x  and 
A 2  are  unequal, 

r      tt    .        t      rr    ,        t      ft  ,         r       //        a  /qk\ 

For     A,  (1,/  1,/'  +  „/  „/'  +  &o.j  =  1,/'  (JjY  +  W  (JJ  +  &c. 

/r/©v     ,v«?@\"    «      >  ,  ,   „      ,  „    o  v 

=  m  (  -7- )  +  m  [~j~j  +  &c.  =  A3  (»?i  m    +  r?2  *?2   +  &o.j. 


We  shall  now  show  that  @  becomes  of  the  required  form. 


Stability  of  Motion.  461 

r    ,    .  (KB        ,  d<&        ,d&  ,d<S 

In  fact  — -  =  ih  —  +  m  —   .   .   .  +  n»  — 

«4i  arii  ch)o  dr\n 

fd<S\'       (d<&V     .        w   , 

"Vl(j7l  +  m\7ri    +  &C<  =  Al  Wl  V1  +  V2V2  •  .  •  +  Tin  Vn) 


=  Ai ! (m2  +  V22  +  >?/2  +  &c)  Z1  +  (m V  +  ihV  +  &o.)  Z%  +  &c. } 

It  can  be  shown  in  a  similar  manner  that 

~jy-  =  A2  (17 1"3  +  if*"8 .  . .  +  W2)  £3,  and  so  on. 
If  then  we  assume,  as  is  allowable, 

»7i     +  Vz    +  Vi     ...+  Tin  .=  L,    rii     +  rjo     +  jjs      •  •  •  +  »y»     =  1,  &0., 

we  have  the  equations 

d<&  d<B  ,     y         n 

•jy  =  Ai4i,    -«r  =  A242,  &c. 

Hence  23  =  X&  +  A2£22  . . .  +  \£n\  (36) 

and  at  the  same  time 

r?!2  +  l|,»  . .  .  +  r?,f  =  V  +  &  •  •  •  +  &\  (37) 

The  constants  r?/,  %',  &e. ;  ij/',  ?/2",  &c,  are  obviously  the 
values  which  #„,  trn,  &c.  ;  r/12,  r/22,  &c.  (Art.  311)  take  in  the 
particular  case  in  which  2  &  is  of  the  form  n*  +  r?22  .  .  .  +  ?jw2. 

As  the  transformation  above  is  real,  it  follows  that  if  @  be 
reduced  in  r/y?y  ?r^//  to  a  form  which  contains  only  the  squares 
of  the  variables,  the  signs  of  the  coefficients  of  the  different 
squares  are  the  same  as  those  of  Ab  A2,  &c. 

In  order  that  every  term  in  the  general  values  of  t,u  £2,  &c. 
should  be  periodic,  it  is  necessary  (Art.  311)  that  all  the  roots 
of  A  =  0  should  be  positive.  This  condition,  as  we  have  just 
seen,  is  fulfilled  if  ©  be  reducible  to  the  sum  of  a  number  of 
squares  with  positive  coefficients — in  other  words,  if  the  func- 
tion V  (Art.  310)  be  a  minimum  for  the  position  of  equilibrium 


462 


Small  Oscillations. 


of  the  system.  In  this  case,  the  system  being  slightly  disturbed 
in  any  manner  from  its  position  of  equilibrium  has  no  ten- 
dency to  depart  far  from  this  position,  and  consequently 
&,  £2,  &c.  must  remain  small  throughout  the  motion.  The 
motion  is  then  stable  in  its  character  whatever  be  the  direc- 
tions of  the  initial  disturbances,  and  the  position  for  which 
£1,  £2,  &c.  are  zero  is  one  of  stable  equilibrium. 

If  V  be  not  a  minimum  for  the  position  of  equilibrium  of 
the  system,  that  is,  if  some  of  the  coefficients  in  @  when 
reduced  to  the  above  form  be  negative,  terms  will  in  general 
occur  in  £1,  £$,  &c.  which  increase  without  limit  with  the  time. 
In  this  case  the  position  is  not  one  of  stable  equilibrium, 
and  the  motion  will  not  consist  of  small  oscillations,  unless 
the  original  disturbances  be  such  that  the  arbitrary  constants 
multiplying  terms  in  £l9  &c,  which  increase  without  limit  with 
the  time,  are  each  zero. 

316.  Case  of  Equal  Roots. — When  the  equation  A  =  0 
has  equal  roots,  the  solution  in  Art.  310  of  the  differential 
equations  (18)  seems  to  fail  from  not  containing  the  requisite 
number  of  arbitrary  constants ;  and  we  might  suppose  that 
terms  containing  t  as  a  factor  would  occur  in  the  values  of 
5i,  £2?  &c.,  and  therefore  that  a  stable  motion  of  oscillation 
would  not  take  place  for  all  possible  small  disturbances. 
Lagrange  and  Laplace  both  fell  into  this  mistake,  which  was 
first  pointed  out  by  Dr.  Eouth. 

The  true  theory  depends  upon  the  circumstance  that  when 
the  equation  A  =  0  has  a  double  root  Xi,  the  system  of  n  linear 
equations  for  determining  ax,  az,  &c,  Art.  311,  are  no  longer 
independent,  but  can  be  satisfied  by  (n  -  2)  of  these  quanti- 
ties, the  remaining  two  being  arbitrary. 

This  may  be  proved  as  follows  : — 


If  we  put 


A! 


-Si 


»12, 


.-    Si 


.     .-So 


,sinj  S2M,  •  •  •  &n     Swn 

where  A19  A2,  .  .  .  An  are  functions  of  X,  we  have 


Equal  Roots. 

463 

dX  " 

dA'  dAi 

dAx  dX 

dA'dA2 

+  dA2   dX  '  '  ' 

tfA' 
dAn 

r/Aw 

(38J 

"we  next 

suppose 

A!=A2  =  A3  .  . 

■  =A„ 

=  A, 

(38) 

becomes 

d& 
dX 

An  +  A22  .  .  .  + 

A„„. 

(39) 

Now,  from  (28)  it  appears  that  when  A  vanishes  An  and 
A 22  have  the  same  sign,  and  this  holds  good  for  any  two  of 
the  determinants  on  the  right-hand  side  of  (39)  ;  but  if  Xi  be 
a  double  root  of  the  equation  A  =  0  the  right-hand  side  of 
(39)  must  vanish  for  this  value  of  A,  and  as  all  its  terms  have 
the  same  sign,  each  must  vanish  separately.  Again,  when 
A  and  An  vanish  it  appears  from  (28)  that  Ai2  must  vanish 
likewise,  and  the  same  is  true  for  every  first  minor  of  A. 

We  conclude  that  when  A  is  a  double  root  of  the  equation 
A  =  0,  the  system  of  n  linear  equations  (19)  of  Art.  311  can 
be  satisfied  by  (n  -  2)  of  the  quantities  ij/,  rj/  .  .  .  nn\  the 
other  two  remaining  arbitrary. 

A  case  of  equal  roots  has  been  already  considered  in 
Art.  308. 

We  can  now  show  that  when  two  roots,  AL  and  A2,  are 
equal,  the  method  already  given  of  effecting  the  orthogonal 
transformation  still  holds  good  with  a  slight  modification. 
In  fact  we  have,  as  before,  Art.  315, 

Vi    *?i      +  172    V2      +  &C.  =  U,      in     17!      +  t],     r/2      +  &c.  =  0, 
&C.  =  0,  &C.  =  0  ; 

but  in  the  present  case  rj/  :  rjS  and  k\"  :  i\"  are  both  arbitrary, 
and  the  two  systems  r?/,  %',  ij3',  &o.  and  ?;",  jj3",  r/3",  &c.  differ 
only  in  consequence  of  different  values  having  been  assigned 
to  these  two  arbitrary  ratios.  By  means  of  one  of  these  ratios 
we  can  now  satisfy  the  single  equation 

>h    v\\     +  m    Vz     •  •  .  +  tin    Tt\n     =  0, 

whilst  the  other  still  remains  arbitrary.  Hence  the  transfor- 
mation is  complete,  but  one  of  the  ratios  which  is  determined 
in  the  case  of  unequal  roots  remains  arbitrary  in  the  case  of 
equal. 


464  Small  Oscillations. 

The  results  obtained  above  for  the  determinants  An,  A22, 
&c.  may  be  extended,  as  in  Art.  314,  to  the  first  minors  of  A 
in  its  most  general  form.  We  may  then  assert,  in  general, 
that  when  A  is  a  double  root  of  the  equation  A  =  0,  the 
system  of  n  linear  equations  (19)  can  be  satisfied  by  (n  -  2) 
of  the  quantities  al9  a2,  .  .  .  an,  the  other  two  remaining 
arbitrary. 

The  conditions  to  be  fulfilled  in  the  case  of  equal  roots 
might  have  been  deduced  at  once  from  the  consideration  that 
the  roots  of  the  equation  An  =  0  separate  those  of  A  =  0,  as 
shown  in  Art  314. 

If,  on  the  other  hand,  some  method  different  from  that 
of  Art.  314  be  adopted  to  prove  the  reality  of  the  roots  of  the 
equation  A  =  0,  then  the  method  of  the  present  Article  may 
be  employed  to  investigate,  as  above,  the  case  of  equal  roots, 
and  also  to  show  that  the  roots  of  the  equation  An  =  0  separate 
those  of  the  equation  A  =  0. 

317.  General  solution  of  the  Differential  Equa- 
tions in  the  case  of  Equal  Roots. — When  the  roots  of 
the  equation  A  =  0  are  all  unequal  and  positive,  equations 
(21)  may  be  written 


(40) 


|i  =  #n  sin  t  VAi  +  flt'n  cos  t  Vai  +  #12  sin  t  VA2  +  #'12  cos  t  Va2  +  &0. 
I2  =  #21  sin  t  Vai  +  rt'21  cos  t  Vai  +  o.2%  sin  t  VA2  +  #'22  cos  t  Va2  +  &c. 
&c.  =  &c. 

where  the  2n  constants  au,  a'u,  a12,  am  &c,  in  the  expression 
for  £1  are  all  arbitrary,  and  the  corresponding  constants  in 
g2,  &c.  may  be  found  in  terms  of  these  arbitrary  constants 
by  the  solution  of  linear  equations,  the  equations  connecting 
an,  #21,  «3i?  •  •  •  an\  being  the  same  as  those  connecting  a'Ui 

a  21,  ft  3i»  •  •  •  (i  m.' 

If  now  two  roots  Ai  and  A2  of  the  equation  A  =  0  become 
equal,  equations  (40)  are  reduced  to  the  form 

£1  =  «n  sin  t\/\i  +  «'n  cos  ^Ai  +  #13  sin  t\/\3  +  a  n  cos  t\/\z  +  &c.\ 

£2  =  #21  sin  t\/^i  +  a  21  cos<-\/ai  +  #23  sin  t\/te  +  a'23  cos  t  V  A3  +  &c.  \..    (41) 

In  -  «n\  sin  t  \/x\  +  a'ni  cos  ty'xi  +  a»3  sin  ty  A3  +  #'»3  cos  t  v  A3  +  &c./ 


Principal  Coordinates  §  Directions  of  Harmonic  Vibration.  465 

In  this  case  there  are  only  2(^-1)  arbitrary  constants  in  gij 
but  since  the  system  of  n  linear  equations  corresponding  to 
At  can  (Art.  316)  be  satisfied  by  (n  -  2)  of  the  unknown  quan- 
tities, the  other  two  remaining  arbitrary,  we  may  in  the 
present  case,  in  addition  to  the  (2n  -  2)  constants  in  £l5  con- 
sider a2l  and  a-n'  also  as  arbitrary.  We  thus  have  still  2n 
arbitrary  constants  altogether,  and  the  solution  of  the  diffe- 
rential equations  (18)  is  therefore  complete.  A  particular 
case  of  this  has  been  already  considered  in  Art.  308.  It  is 
easy  to  see  that  we  may  still,  if  we  please,  express  the  values 
of  ?!,  &c.  by  equations  (21),  but  when  Ax  =  A2  the  constants 
tf2i  and  a22  are  arbitrary,  as  well  as  KXall9  K2al2y  xu  and  x^  and 
in  terms  of  these  six  we  can  express  the  four  arbitrary 
constants  which  belong  to  the  solution  of  the  differential 
equations. 

If  there  be  several  distinct  double  roots  similar  considera- 
tions apply  to  each  of  them,  and  in  general,  corresponding 
to  each  doable  factor  of  A  there  are  four  arbitrary  constants 
in  the  solution  of  the  differential  equations. 

The  preceding  investigation  can  be  readily  extended  to 
the  case  in  which  the  equation  A  =  0  has  r  equal  roots. 

In  this  case  2r  constants  an,  a21,  .  .  .  arl,  an',  a2l',  .  .  .  ar{ 
are  arbitrary,  and  the  n  linear  equations  corresponding  to  the 
multiple  root,  which  in  general  determine  (n  -  1)  quantities 
in  terms  of  the  remaining  one,  are  equivalent  to  only  (n  -  r) 
independent  equations. 

In  fact,  from  what  has  been  proved  above,  it  appears  that 
every  double  root  of  the  equation  A  =  0  must  be  a  root  of 
An  =  0.  Hence  if  the  former  equation  have  r  equal  roots 
the  latter  must  have  (r  - 1).  Again,  it  is  plain  that  An  is 
related  to  An22  in  the  same  way  in  which  A  is  related  to  An, 
and  so  on.  "We  may  therefore  conclude  that  if  the  equation 
A  =  0  have  r  roots  equal  to  Ax,  then  (r  -  1)  successive  minors 
of  A  must  vanish  for  that  value  of  A. 

318.  Principal  Coordinates  and  Directions  of 
Harmonic  Vibration. — Since  in  the  present  case  the 
equations  are  linear  which  connect  different  sets  of  co- 
ordinates, the  generalized  components  of  velocity  are  ex- 
pressed in  terms  of  each  other  by  the  same  equations  as 
those  which  connect  the  corresponding  coordinates.     Hence 

2H 


466  Small  Oscillations. 

the  transformation  of  coordinates  by  which  2&  becomes 
Si2  +  V  . . .  +  In,  reduces  2  T  to  the  form  ^  +  £23 . .  .  &.  Now, 
Art.  315,  2©  is  in  this  case  of  the  form  A^2  +  X2?22 . . .  +  X„?»2, 
and  therefore  by  the  solution  of  the  differential  equations  for 
this  particular  set  of  generalized  coordinates  we  have 

Ci  =  h  sin  *(Va!  +  xi),  &  =  *a  sin  ('  Va8  +  %*),  . . .  f»  =  *«  sin  (*  V\^  +  X»)>  (42) 

where  Xi,  &c.  are  the  roots  of  the  equation  A  =  0,  and 
hi,  k2,  .  .  .  km  xi»  X2»  •  •  •  X»  are  arbitrary  constants,  2w  in 
number. 

The  coordinates  f u  J2,  &c.  are  called  the  Principal  Co- 
ordinates of  the  oscillating  system. 

The  Cartesian  coordinates  a?,  y,  s  of  any  point  of  the 
system  are  given  in  terms  of  the  principal  coordinates  by 
equations  of  the  form 

x  =  x0  +  A£i  +  A£2  .  .  .  +  AnZn  \ 
y  =  y.  +  B£i  +  B&  .  ..  +  £„£»{, 

z  =  s0  +  C\£i  +  C2?2  .  .  .  +  <?„?„ 

where  x0,  yQ,  s0  are  the  values  of  x,  y,  z  respectively  for  the 
position  of  equilibrium,  and  Alt  Blt  d,  A2,  B2,  C2y  &c.  are 
constants  depending  (Arts.  310,  315)  on  the  coefficients 
/u,/12,  &c,  gii,  qi2,  &c,  that  is  on  the  connexions  between  the 
several  particles,  and  on  the  forces  acting  on  the  system. 

From  (42)  it  appears  that  the  motion  of  each  particle  is 
in  general  resolvable  into  n  simple  vibrations  whose  periods 
are 

2tt         2tt  2tt 

v  Ai     *y  X2         v  Xn 

The  motion  of  any  one  particle  being  determined,  that  of  any 
other  consists  of  simple  vibrations  having  the  same  periods, 
i.  e.  harmonic,  with  the  former. 

The  direction  of  motion  for  the  particle  xi/z,  arising  from 

the  simple  vibration  whose  period  is  - — — ,  is  found  by  suppos- 

v  Xi 
ing  £2,  ?3,  •  •  •  £n  to  be  each  zero,  and  depends  upon  the 


Principal  Coordinates.  467 

constants  AXi  Bu  d.  Hence  the  directions  of  the  several 
component  vibrations,  as  well  as  the  ratios  of  their  amplitudes 
for  the  different  particles  in  any  one  harmonic  set,  depend  on 
the  particulars  of  the  system,  i.e.  on  the  connexions  and 
forces ;  and  are  independent  of  the  particulars  of  the  motion, 
i.  e.  of  the  initial  positions  and  velocities. 
The  several  systems  of  directions 

MAft,  A(B{GU  A{'B('C{\  &c), 

(a2b2c2,  a:b:c2\  A:fB:fc2",  &c), 

&G. 

along  the  constituents  of  any  one  of  which  if  the  particles 
xyz9  x'yz,  x'lj'z^  &c.  were  simultaneously  displaced  they 
would  all  vibrate  in  the  same  period  or  harmonically,  are 
termed  the  directions  of  harmonic  vibration. 

The  simple  harmonic  functions  of  the  time  which  occur 
in  the  expressions  for  £1}  &o.  given  by  equations  (21)  differ 
in  general  only  by  constant  multipliers  from  the  values  of 

ZlS    ?2,    &C. 

If  we  put  ki  sin  (t  */\x  +  Xl)  =  ^,  Ka  sin  (t  ^\2  +  X2)  =  ^2, 
&c,  we  may  express  ^1}  ip2,  &c.  in  terms  of  ?i,  $2,  &c,  as 

follows  : — 

Let  2^  =f11a112  +f22a3l2  +f33a31t  +  2f12ana2l  +  2f13a11an  +  &c. 
2«72  "=/ii«i22  +/22«222  +/33«322  +  2fl2a12a22  +  2flzal2ai2  +  &c. 

&U   =  /ll«11^12  +/22«21^22   +/33«,3lrtr32   +/l2(«ll«22  +  «12«2l)   +  &0. 

&c  &c.  &c.  &c. 

and  let  @i,  62,  612,  &c.  denote  the  expressions  obtained  by 
the  substitution  of  qu,  q12,  &c.  for/n,/^  &c.  in  #,  %  £ia,  &c. 
We  have,  then, 

2&i  =  au  —  +  a21  —  .  .  .  +  anl  - — ,  (44) 

dan  da21  danl'  v     ' 

ney  d$rn  d$r%  dST2 

2y2  =  a12  -j—  +  ajB  -r—  . . .  +  a„2  -7—,  (45) 

0#12  "«22  rffl«2  V         ^ 

«7i2  =  «i2  -7—  +  «a  -7—  +  &c.  =  «n  —  +  a2l  — —  +  &c.  ;         46 
dan  achi  dal2  da22 

&c.  &c.  &c.  &c 

2  H  2 


468 


Small  Oscillations. 


and  similar  equations  hold  good  for  ©1?  ©2,  ©12,  &c.  It  is 
easy  to  see  that  ^12,  #i3j  &c.,  @i2,  ©13,  &c.  are  each  zero.  In 
fact  by  (22)  we  have 

Xi  3—  =  -7 — ,      Ai-t-    =  -j— ,     Ar  j—  =  -7—,  &C.       (47) 

from  which  by  multiplication  and  addition  we  get  Xi^w  =  ©12. 
In  like  manner,  Xa£'i2  =  ©12,  and  therefore  in  general  ,9i2  =  0, 
and  ©12  =  0. 

From  (47)  we  have  also  ©x  =  A^i,  and  in  a  similar 
manner  ©2  =  X2^2,  &c  ,,  ^ 

If  now  we  multiply  the  first  of  equations  (21)  by  -— ,  the 

d&  dS?i 

second  bv  — -,  the  third  by  - — ,  and  so  on,  and  add,  all  the 

J  d(h?  d(hi 

simple  harmonic  functions  of  the  time  except  \p1  disappear. 
In  like  manner  we  can  find  \p2j  t//3,  &c.  and  thus  we  obtain 


dau         da  21 


danl 


nrr,      d$rM         d&  d& 

dan         da  2i  danZ 

daln         da2n 


dPn  ~ 

dann 


(48) 


Since  <9"i  is  a  homogeneous  quadratic  function  of  an,  (hi, 
&c,  and  57 the  same  function  of  %l9  g2,  &c.,  it  is  plain  that  the 
first  of  equations  (48)  may  be  written 

d$  d?  d9 

d£i  d&  d%n 

now  from  (21)  we  have  -7^  =  an,    -r^-  =  a2u  &c,  and  there- 
d\pi  dxpi 


fore  we  obtain  2Srnp1 


d$r_ 

#1 


dS? 


In  like  manner  2#>i£«»  =  -77-,  &c ;  hence  we  have 
dxpi 


(49) 


Effect  of  Increase  of  Inertia.  469 

In  a  precisely  similar  manner  we  can  show  that 

@ = @^r- + e,^,2. . . + @„^ns=  \i^i8+  xa#^aa.  •  • + x«#^v  (so) 

If  we  select  the  constants  an,  a12,  a1?„  .  .  .  aln  so  as  to 
satisfy  the  equations  ft  =  1,  &  =  1,  . . .  #»  =  1,  the  simple 
harmonic  functions  ipl9  ip2,  &c.  express  the  values  of  the 
principal  coordinates  of  the  system. 

When  the  harmonic  determinant  equation  has  equal  roots 
the  orthogonal  transformation  which  reduces  (5  to  its  canoni- 
cal form  though  valid  is  no  longer  determinate  Art.  (316), 
and  there  are  an  indefinite  numher  of  sets  of  principal  coordi- 
nates. 

319.  Effect  of  Increase  of  Inertia.— If  the  mass  or 
inertia  of  any  part  of  a  moving  system  be  increased,  the 
expression  for  the  kinetic  energy  receives  thereby  the  addition 
of  one  or  more  terms  of  the  form  vQ\  where  v  is  a  positive 
constant,  and  6  is  a  linear  function  of  the  generalized  com- 
ponents of  velocity.  The  coordinates  may  be  transformed 
in  such  a  way  as  to  make  the  linear  functions  0,  &c.  identical 
with  an  equal  number  of  the  generalized  coordinates  \X9  &c. 

If  the  forces  acting  on  the  system  remain  unaltered,  and 
if  there  be  only  one  additional  term  in  the  expression  for  the 
kinetic  energy,  the  harmonic  determinant  A'  of  the  system 
in  which  there  has  been  an  increase  of  mass  or  inertia,  is 
given  then  by  the  equation 

X(/n  +  v)  -  qll9     X/12  -  qu     • 

X/l2  -  012 


A'  = 


A  +  v\  An, 


where  A  is  the  harmonic  determinant  of  the  original  system. 
If  the  original  position  be  one  of  stable  equilibrium  all 
the  roots  Xi,  .  .  .  \n  of  the  equation  A  =  0  are  positive,  and 
are  separated  by  the  roots  fii9  .  .  .  fin-\  of  the  equation  An  =  0. 
Hence  Ar  is  positive  for  X  =  Xi,  negative  for  X  =  /ui,  negative 
for  X  =  X2,  positive  for  X  =  ju2,  and  so  on.  Consequently  the 
roots  of  the  equation  A'  =  0  are  each  less  than  the  correspond- 
ing root  of  the  equation  A  =  0,  but  are  all  positive  and  are 
separated  by  the  roots  of  the  equation  An  =  0. 


470  Small  Oscillations. 

It  follows  from  what  has  been  said,  that  when  the  forces 
remain  unaltered  an  increase  of  mass  increases  the  several 
periods  of  vibration. 

If  the  generalized  coordinate  0  or  &  were  rendered 
invariable  the  system  would  have  only  [n  -  1)  degrees  of 
freedom,  and  the  harmonic  determinant  would  become  An. 
Hence  no  root  of  the  equation  A  =  0  is  diminished  by  an 
increase  of  inertia  as  much  as  it  would  be  by  rendering  the 
corresponding  coordinate  invariable. 

It  follows  that  if  any  period  of  oscillation  belong  to  a 
system  both  before  and  after  a  certain  coordinate  has  been 
rendered  invariable  this  period  belongs  also  to  the  system 
when  the  mass  corresponding  to  this  coordinate  is  increased. 

The  substance  of  this  Article  is  taken  from  Eouth's 
Rigid  Dynamics. 

320.    Energy  of  an  Oscillating  System. — If  we  put 

t  ^/Ai   +    Xl  =    0U     t  \/^2   +    X2   =    02,      &C, 

and  substitute  in  T  the  values  of  £i,  £2,  &c.  obtained  by  dif- 
ferentiating equations  (42)  we  have 

2T  =  XJcS  cos2  0!  +  A2&22  cos2  02  +  &c.  (51) 

Again,  substituting  in  V  the  values  of  &,  £2,  &c  we  have 

2  V  =  2  Vo  +  Xifc"  sin2  fa  +  A2&22  sin2  <p2  +  &o.  (52) 

Hence,      2(T+  V)  =  2F0  +  XA1  +  XA2  .  .  .  +  Xjkn\       (53) 

From  equations  (51)  and  (52)  it  is  plain  that  the  sum  of  the 
kinetic  and  potential  energies  corresponding  to  each  oscilla- 
tion is  constant  at  each  instant  and  equal  to  double  the  mean 
value  of  either. 

It  is  plain  also  that  the  mean  value  of  the  total  kinetic 
energy  is  equal  to  that  of  the  total  potential  energy  due  to 
the  oscillatory  motion. 

The  general  expression  for  the  kinetic  energy  is  found  by 

substituting  \px  for  \pi,  \p2  for  fa  &o.  in  (49) . 
We  have  thus 

T  =  Xxici2  &x  cos20!  +  X2K22  &*  cos202 . . .  +  Anic„2^;  cos2#n.  (54) 
From  (50)  we  obtain 
V=  V9 +Xi  K!2  &\  sin20!  4  X3  k22  #j  sin2  02 . .  .  +  X„  k712  S?n  sin20„.  (55) 


Examples. 


471 


Examples. 

In  the  following  examples  the  small  oscillations  of  the  system  are  to  be 
determined  in  each  case  : — 

1.  A  number  of  balls  suspended  by  a  fine  cord  hang  in     

a  vertical  line,  and  are  slightly  displaced  in  the  same  vertical 
plane. 

Let  xi,  z\  ;  x«,  z2,  &c.  be  the  horizontal  and  vertical 
coordinates  of  the  balls  ;  «i,  ao,  &c.  the  distances  from 
the  point  of  suspension  to  the  first  bail,  from  the  first  ball 
to  the  second,  and  so  on  ;  Q\,  02,  &c  the  angles  which  «i, 
fl-2,  &c.  make  with  the  vertical  at  any  instant.  The  weight 
of  the  cord  being  neglected,  the  distances  «i,  &c.  are  inva- 
riable ;  then 


X2  =  a\B\  +  a-x  62, 


=  01  cos  0i  =  fli(l  -i6r), 
ai(l-£0i2)  +  «2(l-£022),  &c. 


Substituting  in  the  general  dynamical  equation,  neglecting 

bi,   &c.  and  equating  to  zero  the  coefficients  of  50i,  502,  &c.  we  have,  after 

dividing  the  first  equation  by  «i,  the  second  by  a2,  &c. 

(mi  +  m%  +  »»3  +  &c.)  fli^'i  +  [nn  +  m$  +  &c. )  «2  62  +  (wa  +  &c.)  03  03  +  &c. 

+  (mi+m2+&c.)ffdi  =  0, 

{m%  +  W3  +  &c.)  a\  d\  +  (m2  +  ni5  +  &c.)  «2  02  +  (m3  +  &c.)  a2  03  +  &c. 

+  (W2  +  &C.)  003  =  0, 

mna\9i  +  wna202  +  mnOi'fh  +  &c.  +  mnan6n  +  WngQn  =  0. 
Hence,  assuming 

0i  =  ha  sin  [t  Va  +  x)>     0*  =  #0  sin  (t  Va  +  x),  &c. 

where  A-  and  x  are  arbitrary  constants,  we  get  to  determine  a,  )8,  7,  &c,  and  \ 

the  equations 

(?«i  +  w2  +  &c.)(aiA-5')o  +  (w2  +  W3  +  &c.)a2A)8+  (w3  +  &c.)a3A7+&c.  =  0, 

(W2  +  W3  +&c.)aiAa+  (»»2  +  »«3  +  &c.)(«2A.-5')i8+  (mz  +  &c.)  a3\y  +  &c.  =  0. 

(flio  +  «2)8  +  «37  +  &c.  +  flnw)  A  -  ga>  =  0. 

This  problem  can  also  be  treated  by  the  general  method  of  Art.  310.  For, 
since  the  vertical  motion  of  each  ball  is  very  small  in  comparison  with  its  hori- 
zontal motion,  the  velocities  si,  Z2f  &c.  may  be  neglected  ;  and  we  readily  find 

2T-  mi  <7r  0r  +  w2  (tfi  0i  +  a2  02)2  +  mz  (ax  0i  +  a2  02  +  «3  03) 2 
.  .  .  +  m»  (ai  01  +  «2  02  +  •  •  •  +  dn  0„)2. 

Also,  if  the  potential  energy  be  estimated  from  the  position  of  equilibiium  of 
the  system, 

2  V  =  mi  ga\  0r  +  W20  («i  Or  +  «2  022)  +  . . .  +  »M  (ai  #i2  +  #2  022  +  •  •  •  +  ««  0n2)  • 

The  preceding  differential  equations  immediately  follow  from  these  equa- 
tions by  the  method  of  Art.  310. 


472  Small  Oscillations. 

2.  The  system  of  balls  suspended  as  in  the  last  example  are  displaced  in 
different  vertical  planes. 

In  this  case,  0i  and  cpi  being  the  angular  displacements  of  «i  towards  the 
axes  of  y  and  x,  B2  and  <p2  those  of  a2,  &c. 

x\  =  a\<p\,    yi  =  ai9i,     zi  =  fli (1  -  i0i2  -  |^)r),  &c. ; 

then  0i,  02,  &c.  are  independent  of  (pi,  (p2,  &c.  as  in  Ex.  1,  Art.  308.  The 
values  of  0i,  &c.  are  therefore  the  same  as  in  the  last  example,  whilst  those  of 
0i,  &c.  differ  from  them  only  by  having  a  different  set  of  arbitrary  constants. 

3.  A  number  of  rigid  bars  are  hinged  together  in  the  same  vertical  line, 
and  are  displaced  in  the  same  vertical  plane. 

If  x\,  z\  be  the  coordinates  of  any  point  of  the  first  bar,  x2,  z2  those  of  any 
point  of  the  second,  &c.  and  0i,  02,  &c.  the  inclinations  of  the  bars  to  the 
vertical ;  x\  =  rtfi,  z\  =  n  (1  -  |0r),  x2  =  «i0i  +  r202,  z*  =  «i  (1  -  \Br) 
+  r2  (1  -  i022),  &c. 

Proceeding  as  in  Ex.  1,  we  obtain 

{mih2+  (m2  +  m-i  +  &c.)  «i2}  B\  4  {m%h  +  («»3  +  &c.)  a2}  aiB\ 

+  {m3b3  +  (mi  +  &c.)  a3}  a{B3  +  &c.  +  {niih  +  (*«2  +  m%  +  &c.)  «i}  ^0i  =  0, 

{m2b2  +  (m3  +  &c.)  a2}  «i0i  +  {m2k22  +  (m3  4-  &c.)  «22}  B2 

+  {m3b3  +  (mi  +  &c.)  a3}  a2B3  +  &c.  +  {m2b2  +  (m3  +  mi  +  &c.)  a2}  g62  =  0, 

Wnbndi'Bi  +  mnbna292  +  mnbna3B3  +  &c.  +  mnktfdn  +  mnbngBn  =  0, 

where  k\y  Jc2,  &c.  are  the  radii  of  gyration  of  the  bars  round  their  extremities, 
and  #i,  b2,  &c.  the  distances  from  their  extremities  to  their  centres  of  inertia. 
The  solution  of  these  equations  is  obtained  in  the  same  manner  as  in  Ex.  1 . 

4.  Two  balls,  m\  and  m2,  are  suspended  by 
cords,  whose  lengths  are  a\  and  a2,  from  the  extre- 
mities of  a  bar  whose  position  of  equilibrium  is 
horizontal,  and  which  hangs  from  a  fixed  point  by 
another  bar  c  rigidly  attached  to  the  former  at  a 
point  whose  distances  from  the  extremities  of  the 
horizontal  bar  are  b\  and  b2.  The  system  is  dis- 
placed in  the  vertical  plane  which  it  occupies  when 
in  equilibrium. 

Let  0i,  02,  03,  be  the  inclinations  of  a\y  a2,  andc 
to  the  vertical ;  x\,  z\y  and  x2,  z2  the  coordinates  of  m\  and  m2  ;  then 

xi  =  0101  +  h(l-  \Br)  +  cB3,     «i  =  *i  (1  -  W)  -  Ms  +  o  (1  -  |032), 

x2  =  a2B2  -b2{\-  |032)  +  cBz,     z2  =  <h  (1  -  \B22)  +  b2Bz  +  e(l  -  |032). 

Also  let  d\  and  d2  be  the  distances  from  0  to  the  extremities  of  the  lever. 
Then,  neglecting  small  quantities  beyond  the  second  order,  we  readily  get 

2T=  mi  (ai20i2  4-  2aic8\&3  +  rfi2032)  +  m2  (a22B22  +  2a2cB2B3  +  d22B32)  +  m3k32B32, 

where  W3&32  is  the  moment  of  inertia  of  the  lever  round  the  point  0. 
Also,  if  th  weight  of  the  bar  c  be  neglected, 

2  V  =  m\ga\Bi2  +  m2ga2B22  +  MgcBz2, 
where  M  =  m\  +  m2  +  m3. 


Examples.  473 

Hence  the  differential  equations  for  small  oscillatory  motion  are 

(ii'di  +  003  +  gdi  =  0,     a262  +  cd3  +  gd2  =  0, 

and  m\a\B\  +  m2a2B%  -f  Maz'Oz  +  ilf<703  =  0, 

where  m\d^  +  m2d22  +  nizfo2  =  Maze. 

If  we  now  assume 

0i  =  asin  (*^+x),      02  =  /3fiin(^^H-x^     03  =  7.sin  (<^  +  x), 

it  is  readily  seen  that  A  is  a  root  of  the  cubic 

M (a  -  «r)  (A  -  fl-o)  (A  -az)  —  K («*i«i  +  w^ofc)  c  +  (mi  +  m3)  axa2c  =  0. 

It  should  he  noted  that  if  the  cords  are  equal  in  length,  i.  e.  a\  =  a2,  then  a\ 
is  a  root  of  this  cubic,  and  the  remaining  roots  are  given  by  the  quadratic 

M{\  -  a\)  (A  —  as)  —  (mi  +  mo)  die  =  0. 

This  latter  furnishes  the  solution  of  the  small  motion  of  a  beam  and  scales, 
oscillating  in  a  vertical  plane  passing  through  the  beam.  [See  Camb.  Math. 
Journal,  vol.  ii.,  p.  120.] 

5.  The  balls  and  cords  in  the  preceding  example  are  replaced  by  two  bars 
which  hang  freely  from  the  ends  of  the  lever. 

Let  h  and  l2  be  the  distances  from  the  ends  of  the  arms  of  the  lever  to  the 
centres  of  inertia  of  the  bars  a\  and  a2j  respectively  ;  and  let  miki2  be  the 
moment  of  inertia  of  the  bar  ax  round  its  upper  extremity,  and  m2k2z  the  corre- 
sponding quantity  for  a2  ;  then  we  readily  find 

2T=  mi  {h20iz  +  2hc6\e3  +  ^i°~032)  +  m  (WW  +  2l2c0293  +  d2Hr)  +  msfeW, 

and,  making  the  potential  energy  zero  in  the  position  of   equilibrium,   and 
neglecting  the  mass  of  c,  we  have 

2V=  mighOi2  +  m2gl2922  +  Mgcdz2. 

Proceeding  as  in  last  example,  and  putting 

.        h2         .        W 
hi  =—,       h2  =  —, 
n  h 

we  get,  for  the  determination  of  A,  the  cubic 

M(\  -  hi)  (a  -  A2)  (A  -  as)  —  Ac  (niih  +  m2l2)  -f  c  (milji%  +  m2l2hi)  =  0. 

As  in  the  last  example,  if  h\  =  h2,  then  hi  is  one  value  of  A,  and  the  other 
roots  of  the  cubic  are  those  of  the  quadratic 

-If  (A.  -  hi)  (\  -  a2)  -  c  (mJi  +  m2l2)  =  0. 

6.  A  rigid  body,  having  a  fixed  point  and  in  stable  equilibrium  under  the 
action  of  a  conservative  system  of  forces,  is  slightly  disturbed. 

Let  the  axis,  a  rotation  round  which  would  bring  the  body  from  its  position 
of  equilibrium  to  its  actual  position  at  any  time  (Art.  249),  make  angles  with 
the  principal  axes  of  the  body  at  the  fixed  point  whose  direction  cosines  are 
I,  m,  n.     Let  <r  be  the  magnitude  of  the  required  rotation,   which  by  hypo- 


474  Small  Oscillations. 

thesis  is  a  small  quantity.  The  coordinates  of  each  point  of  the  hody  are 
then  at  any  time  functions  of  constants  and  of  the  variables,  <t,  I,  m,  n,  or,  of 
the  three  independent  variables  al,  am,  an,  which  may  be  denoted  by  0,  <p,  \p. 

Hence,  as  0,  <p,  i/>  are  small,  and  the  initial  position  is  one  of  equilibrium 
(Art.  310), 

V  =  Vo  +  |  (?1103  +  <?22<£2  +  ?33*P  +   2qn9<t>  +  2^130^  +  2^23^)- 

Again,  neglecting  small  quantities  of  the  second  order,  «i  =  0,  u>z  =  <£,  «3  =  rp ; 
and  therefore  (Art.  263) 

T=i(^02  +  £(p2+  Cty2), 

neglecting  small  quantities  of  the  third  order. 
Hence  equations  (18)  become 

Ad  +  qn6  +  Qi2<p  +  Qiz^p  =  0, 
B<p  +  que  +  q-x<p  +  ?23t//  =  0, 

Cty   +  qnO  +  ?23^>  +  533^  =  0. 

Assuming 

0  =  Jca  sin  (*  Va  +  x)i     <P  =  %&  sin  (*  V\  +  x)>     ^  =  &7  sin  (*  l7*  +  x), 

we  have,  for  the  detennination  of  a,  £,  7,  A,  the  equations 

AXa  =  g-no  +  512)8  +  ?i37> 

-#Aj8  =  £i2a  +  522)8  +  5-237, 

C\y  =  qua  +  523)8  +  5337- 

If  01,  02,  &c.  be  the  values  of  a,  &c.  corresponding  to  Ai  and  A2,  two  of 
the  roots  of  the  cubic  for  A,  it  is  easy  to  see  that 

(Ai  -  A2)  {Aaiaz  +  Bfafa  +  C7172)  =  0  ; 

hence  Actio.*  +  Bfiifiz  +  C7172  =  0, 

and  therefore  also 

02  (51101  +  512)81  +  51371)  +  )82  (5i2«i  +  522^1  +  52371) 

+  72  (51301  +  523)81  +  5337O  =  °- 

Accordingly  the  lines  whose  direction  cosines  are  proportional  to  01,  £1,  71 ; 
02,  /32,  72;  03,  £3,  73 ;  are  conjugate  diameters  of  the  momental  ellipsoid,  and 
likewise  of  the  quadric  E,  whose  equation  referred  to  the  principal  axes  of  the 
body  at  the  fixed  point  is 

511  #-  +  522^  +  533Z2  +  2512  xy  +  2qi3xz  +  2q2*yz  =  K. 

Since  the  initial  position  is  one  of  stable  equilibrium,  E  must  be  an  ellipsoid 
(Art.  315). 


Examples. 


475 


An  angular  displacement  <r\  from  the  position  of  equilibrium  brings  the  body 
into  a  position   whose  potential   energy,    relative   to  the   initial   position,  is 

n  being  the  semi-diameter  of  E  round  which  the  rotation  <n  is 


effected.    Hence  all  small  angular  displacements, 


•hich  are  proportional  to  the 
diameters  of  E  round  which  they  are  effected,  bring  the  body  into  positions  having 
the  same  potential  energy.  On  this  account  Sir  Robert  Ball  calls  E  the  ellipsoid 
of  equal  energy,  or,  the  potential  ellipsoid  corresponding  to  the  initial  position  of 
equilibrium.  The  results  arrived  at  may  be  stated  as  follows  : — The  harmonic 
axes  of  a  rigid  body,  having  a  fixed  point  and  in  stable  equilibrium  under  the 
action  of  a  conservative  system  of  forces,  are  the  three  conjugate  diameters 
common  to  the  momental  and  the  potential  ellipsoids.  This  theorem  is  due  to 
Sir  Robert  Ball. 

7.  If  gravity  be  the  only  force  acting 
on  the  body  in  the  last  example,  show  that 
the  potential  ellipsoid  becomes  a  circular 
cylinder,  and  determine  the  positions  of  the 
harmonic  axes. 

Let  0  be  the  fixed  point,  G  the  initial 
position  of  the  centre  of  inertia,  G'  its  posi- 
tion resulting  from  a  small  angular  displace- 
ment <r  of  the  body  round  the  line  OR  whose 
direction  cosines  are  I,  m,  n.  Draw  GP 
and  G'P  perpendicular  to  OR,  and  G'H 
perpendicular  to  OG.  Then  <r  =  L  GPG', 
and  the  potential  energy  due  to  the  displace- 
ment is  3Rg  .  GPL.  Putting  OG  =  a,  and 
L  GOP  =  p,  we  have 


GE- 


GG'2 
~2a~ 


PG*.ff' 


2a 


=  -  (t-  sin-  p. 
2 


Again,  if  \,  p.,  v  be  the  direction  cosines  of  OG, 

a  cos  p  =  <r  [l\  +  m/x  +  nv)  =  8\  +  (pp.  +  $v. 
Hence,  the  potential  energy  due  to  the  displacement  is 
|3%a  {02  +  <p2  +  \p-  -  (6\  +  (pix  +  M2}, 
and  the  quadric  E  is  determined  by  the  equation 

x2  +  y2  +  z2  -  (\x  +  p.y  +  vz)2  =  K, 

which  represents  a  right  cylinder  having  OG  for  its  axis. 

This  can  also  be  easily  seen  directly  as  follows  : — The  amount  of  energy  T 
required  to  turn  the  body  through  an  angle  a  round  any  semi-diameter  r  of  E  is, 

as  we  have  seen,  \K  —.     Now,  if  r  be  vertical,  T=  0,  and  therefore  r  =  oo  . 

Also,  if  r  be  horizontal,  T  =  Wtyh  (where  h  is  the  height  through  which  the 


476  Small  Oscillations. 

centre  of  inertia  is  raised),  and  is  therefore  constant  if  <r  be  constant,  hence 
r  is  constant :  accordingly  the  corresponding  section  of  E  is  a  circle. 

To  determine  the  harmonic  axes — One  is  the  vertical,  OG,  the  other  two  are 
found  as  follows  : — Draw  the  diametral  plane  of  the  momental  ellipsoid  which 
is  conjugate  to  a  vertical  line  through  the  fixed  point ;  it  will  meet  the  cylinder 
E  and  the  momental  ellipsoid  in  two  ellipses  ;  the  pair  of  conjugate  diameters 
common  to  these  two  are  the  lines  required.  Also,  since  a  horizontal  section  of 
E  is  a  circle,  the  projections  on  any  horizontal  plane  of  the  two  non-vertical 
harmonic  axes  are  lines  at  right  angles  to  each  other.  If  the  body  be  displaced 
without  initial  velocity  there  will  be  no  oscillation  round  the  vertical  axis  ;  and 
if  the  periods  of  vibration  round  the  other  two  axes  be  different,  the  instanta- 
neous axis  of  rotation  will  either  oscillate  or  revolve  continuously  in  the  plane  of 
the  non- vertical  harmonic  axes.  If  after  displacement  an  initial  velocity  be 
imparted  to  the  body,  in  order  that  there  should  be  small  oscillations,  this  initial 
velocity  of  rotation  must  be  round  an  axis  in  the  plane  of  the  two  non- vertical 
harmonic  axes. 

8.  If  the  system  of  forces  acting  on  the  body  be  constant  in  magnitude  and 
direction,  determine  the  values  of  qu,  &c.  in  Example  6. 

Let  X,  Y,  Z  be  the  components  of  the  constant  force  acting  at  any  point  of 
the  body  parallel  to  the  initial  direction  of  its  principal  axes,  and  let  x,  y,  z  be 
the  coordinates  of  the  point  of  application  referred  to  space  axes  coinciding  with 
these  initial  directions,  its  coordinates  referred  to  the  principal  axes  themselves 
being  £,  7),  £.  Then,  if  Gx,  Gy,  Gz  be  the  moments  of  the  forces  round  the 
space  axes,  neglecting  small  quantities  of  the  second  order,  we  have 

Gx  =  2  {(yZ-  zY)  =  2  { („  -  0  +  ft)  Z  -  (C  -  &  +  v0)  Y} 

=  2{VZ-  £Y)  -  02  (tjF+  (Z)  +  <p^Y+  ^%\Z 

=  <r  {-  i2(nY+  cz)  +  w2|r+  tasz}, 

since  2  {t\Z  -  (Y)  =  0. 

The  work  done  by  the  forces  in  turning  the  body  through  d<r  is 
{IGx  +  mGy  +  nGz)  da. 

If  we  substitute  for  Gx  its  value  given  above,  and  make  similar  substitutions  for 
Gy  and  Gz,  we  obtain,  by  integration,  the  terms  of  the  second  order  in  —  V.  Hence 
we  have 

qn  =  2  (nY+  (Z),     2^2  =  -  2  (|F+  rjX),  &c. 

9.  If  a  system  whose  position  is  determined  by  three  independent  variables 
perform  small  oscillations,  prove  that  the  harmonic  axes  are  the  three  conjugate 
diameters  common  to  the  quadrics  whose  equations  are  §f  =  constant,  @  =  con- 
stant (Art.  313). 


(     477 


CHAPTEE    XIV. 

THERMODYNAMICS. 

321.  Mechanical  Equivalent  of  Heat. — The  experiments 
of  Joule  and  others  have  shown  that  whenever  sensible  kinetic 
energy  disappears  without  a  corresponding  increase  of  poten- 
tial energy,  an  amount  of  heat  is  produced  proportional  to  the 
quantity  of  sensible  kinetic  energy  which  has  disappeared. 

A  similar  result  takes  place  in  all  cases  in  which  work  is 
expended  without  producing  a  corresponding  increase  of 
energy ;  and,  conversely,  a  definite  amount  of  heat  can  be 
transformed  into  a  definite  amount  of  work. 

The  number  of  units  of  work  which  the  unit  of  heat  can 
perform  is  called  the  mechanical  equivalent  of  heat,  and  may  be 
designated  by  the  letter  J.  If  the  quantity  of  heat  required 
to  raise  the  temperature  of  the  unit  mass  of  water  from 
0°  to  1°  Centigrade  be  taken  as  the  unit  of  heat,  and  the 
amount  of  work  expended  in  lifting  the  unit  mass  through 
a  height  of  one  metre  as  the  unit  of  work,  the  value  of  J  is 
found  to  be  424.  In  English  units,  i.  e.  if  a  foot  be  taken  as 
the  unit  of  length,  and  temperature  be  estimated  by  Fahren- 
heit's thermometric  scale,  the  value  of  J"  is  772. 

If  Q  be  the  number  of  units  of  heat  imparted  to  a  body ; 
U  its  total  energy,  kinetic  and  potential ;  W  the  work  done 
by  the  body  against  external  forces ;  and  A  Q,  &c.  the  in- 
crements of  these  quantities  at  any  time  reckoned  from  the 
same  instant,  the  experiments  of  Joule,  already  mentioned, 
conduct  to  the  equation 

JAQ  =  AU+AW.  (1) 

If  we  desire  to  give  this  equation  a  purely  theoretical 
basis,  we  have  only  to  assume  that  heat  is  energy  resulting 
from  molecular  motion,  and  that  the  principle  of  the  con- 
servation of  energy  holds  good. 


478  Thermodynamics. 

If  we  seek  to  determine  J  from  Equation  (1)  by  measur- 
ing the  amount  of  AQ,  &c.  in  any -particular  case,  we  are  met 
by  the  difficulty  that  the  value  of  A  TJ  is  in  general  unknown. 
This  difficulty  can  be  got  over  by  bringing  back  the  body 
which  is  being  experimented  on  to  its  initial  condition  :  A  U 
is  then  zero,  and  we  have     J  A  Q  =  A  W. 

When  J  is  known,  heat  can  be  expressed  in  work  units  ; 
and  if  this  mode  of  expressing  Q  be  adopted,  (1)  takes  the 
simpler  form 

AQ=AU+AW.  (2) 

322.  Equation  of  Energy.  —  The  equation  of  the 
preceding  Article  is  one  of  the  two  fundamental  equations 
of  Thermodynamics,  and  may  be  called  the  Equation  of 
Energy. 

In  the  application  of  this  equation  the  substance  under 
consideration  is  in  general  supposed  to  pass  continuously 
from  one  state  to  another,  in  consequence  of  changes  in  its 
temperature  and  in  the  pressure  which  the  unit  area  of  its 
surface  exerts  against  the  surrounding  medium,  this  pressure 
being  supposed  the  same  at  all  points  of  the  surface. 

In  Thermodynamics  we  are  not  usually  concerned  with 
the  kinetic  energy  of  sensible  motion.  In  fact,  the  apparent 
effect  of  heat  on  a  substance  is  either  to  raise  its  temperature, 
or  to  change  its  condition,  or  to  cause  it  to  do  external  work. 
Hence  the  body  is,  in  general,  supposed  to  be  at  rest  in  the 
ordinary  sense,  and  its  kinetic  energy  is  exhibited  in  the 
form  not  of  sensible  motion,  but  of  those  molecular  motions 
on  which  temperature  depends.  This  being  understood,  we 
see  that  the  total  energy  of  a  unit  mass  of  the  substance  at 
any  time  is  a  function  of  two  independent  variables — the 
temperature  t,  and  the  pressure  p. 

As  the  volume  v  of  the  unit  of  mass  depends  on  p  and  t, 
we  may  take  as  independent  variables  any  two  of  the  three 
quantities  t,  p,  and  v. 

If  the  work  done  by  the  body  against  external  forces  be, 
as  is  usually  the  case,  the  work  which  it  does  in  consequence 
of  its  expansion  against  the  pressure  on  its  surface,  and  if 
we  consider  merely  the  unit  of  mass,  it  is  easily  seen  that 


Specific  Heat.  479 

dW  =  pdc,  and  hence  we  have  from  (1)  the  equation 

JdQ  =  ^  dt  +  C^-dv  +  pdv.  (3) 

dt  dv 

323.  Specific  Heat. — The  number  of  units  of  heat 
which  must  be  imparted  to  the  unit  of  mass  of  a  homogeneous 
substance  to  raise  its  temperature  one  degree  is  called  its 

specific  heat,  and  is  equal  to  the  limit  of  — . 

The  specific  heat  in  general  depends  on  the  external  work 
accomplished  by  the  body,  and  is  indeterminate  unless  the 
relation  between  the  variations  of  the  independent  variables 
be  assigned.  If,  however,  the  temperature  of  the  body  be 
raised  either  under  the  condition  that  its  volume  remains 
constant,  or  under  the  condition  that  the  pressure  on  its 
surface  remains  constant,  the  specific  heat  is  a  definite  func- 
tion of  the  variables  on  which  the  state  of  the  body  depends. 

The  specific  heat  at  constant  volume  may  be  designated 
by  Cv,  and  that  at  constant  pressure  by  Cp.  We  have  then, 
from  (3), 

rn     (dU\         rn      (dU\         dv  us 

where  (  -—  }  indicates  the  differential  coefficient  of  JJ  with 
\dt  Jv 

respect  to  t  under  the   hypothesis  that  v  is  constant,  and 

dU\ 

—  1  has  a  similar  signification  in  reference  to  pf  and  where 

dt  Jp 

in  the  equation  for  Cp  we  regard  v  as  a  function  of  p  and  t. 

For  practical  purposes,  when  great  accuracy  is  not  re- 
quired, no  distinction  is  made  in  the  case  of  solid  and  liquid 
bodies  between  the  two  specific  heats,  and  the  specific  heat 
for  each  body  is  assumed  to  be  an  absolute  constant. 


J 


480  Thermodynamics. 


Examples. 

1.  A  raindrop  falls  to  the  ground  from  a  height  of  1272  metres ;  determine 
hy  how  much  its  temperature  is  raised,  assuming  that  it  imparts  no  heat  to  the 
air  or  to  the  ground.  -^ns.  3°C. 

2.  Find  how  much  heat  is  disengaged  if  a  bullet  weighing  50  grammes  and 
having  a  velocity  of  50  metres  per  second  strikes  a  target,  assuming  g  to  he 
9-8  metres  per  second. 

Am.  An  amount  of  heat  sufficient  to  raise  one  gramme  of  water  through 
15°C. 

3.  Supposing  the  Earth  to  have  been  originally  a  nebulous  mass  dissipated 
through  space  ;  find  the  heat  produced  by  its  condensation. 

If  ^be  the  kinetic  energy  generated  by  the  coming  together  of  the  nebulous 

mass,  we  have,  by  Ex.  12,  Art.  138,  $f  =  -=  — .      The  equivalent  amount  of 

Q~     3  urn  mr      3  mar       _,  ,    ...   . 
heat  Q  is  given  by  the  equation  Q=  'jr  =  -  ^  -j-=  ^  — .      Now,  substitut- 
ing for  r  its  value  in  metres  — - ,  and  424  for  J,  we  obtain  Q  =  9000w^, 

approximately.  Hence  the  quantity  of  heat  generated  by  the  condensation  of 
the  Earth  is  90  times  the  amount  required  to  raise  an  equal  mass  of  water  from 
0°  to  100°C. 

4.  Find  the  amount  of  heat  generated  by  the  condensation  of  the  sun. 
Let  Q'  be  the  amount  of  heat  required,  then  M  and  R  being  the  mass  and 

Q'      M*    r      M    r  M  M 

radius  of  the  sun,  we  have,  from  Ex.  3,  -  =  -^ .-  =  -.  ^  -.     Now,  -  = 

324000,  and —=  108,  approximately.      Hence  —  =  3000— ,  nearly.      Again, 

9000  x  3000  =  27000000,  consequently  the  heat  generated  by  the  condensation 
of  the  sum  is  270,000  times  the  amount  required  to  raise  the  temperature  of 
an  equal  mass  of  water  from  0°  to  100°C. 

5.  If  the  sum  be  contracting  in  consequence  of  its  own  attraction  ;  determine 
the  annual  contraction  which  is  required  to  maintain  its  temperature  constant. 

As  in  Ex.  3,  we  have  QR  =  f  ~r-,  and  therefore  -—  +  —  =  0. 

J  u        Jti 

By  observing  the  quantity  of  heat  received  from  the  sun  in  a  given  time  by 

a  given  area  on  the  surface  of  the  Earth,  it  is  easy  to  determine  the  whole 

amount  of  heat  emitted  by  the  sun  in  one  year.     From  this,  and  the  mass  of 

the  sun,  we  can  ascertain  that  the  temperature  of  an  equal  mass  of  water  would 

be  lowered  a  little  more  than  2°  by  losing  this  amount  of  heat.     Consequently, 

1®  = ,  approximately,  and  therefore  to  maintain  its  present  tempera- 

Q       13000000'    F*  J  ..... 

ture  the  sun  should  contract  each  year  by  an  amount  sufficient  to  diminish  its 

diameterby  130^000  °fitSlength- 


Perfect  Gas.  481 

324.  Perfect  Gas. — In  the  case  of  a  perfect  gas  the 
volume  v  of  the  unit  of  mass  is  connected  with  the  pressure  p 
and  the  temperature  t  by  an  equation  of  the  form 

vp  =  v0po  (1  +  at),  (5) 

where  v0  is  the  volume  corresponding  to  the  pressure  p0  and 
the  temperature  zero,  and  a  is  a  constant  which  is  the  same 
for  all  gases,  its  value  being  -^--j  when  temperatures  are 
counted  on  the  Centigrade  thermometer.  If  the  zero  of 
temperature  be  taken  at  -  273°  C,  and  temperature  reckoned 
from  this  origin  be  denoted  by  T,  we  have,  putting  R  for 

aV0po, 

vp  =  RT.  (6) 

The  experiments  of  Joule  and  Thomson  have  shown  that 
if  the  volume  of  a  gas  vary  without  any  heat  being  imparted 
or  abstracted,  the  temperature  remains  constant,  provided  no 
external  work  is  done.  If  now  in  (2)  we  make  AQ  =  0  and 
/\  W  =  0,  we  have  A  U  =  0  ;  hence  it  appears,  that  if  the 
temperature  of  a  gas  remains  invariable  so  likewise  does  the 
internal  energy,  which  is  therefore  a  function  of  the  tempe- 
rature alone.     In  this  case,  by  (4)  and  (6),  we  have 

JCP  =  JCV  +  p  (j\  =  JCV  +  R.  (7) 

The  experiments  of  Eegnault  have  shown  that  the  specific 
heat  of  a  gas  at  constant  pressure  is  independent  of  the  pres- 
sure, being  a  constant  for  each  gas.  From  this  it  follows 
by  (7)  that  the  specific  heat  at  constant  volume  is  likewise  a 
constant.  In  the  case  of  a  perfect  gas  equation  (3)  accord- 
ingly becomes 

JdQ  =  JCvdT  +  pdv.  (8) 

If  Q  be  the  heat  imparted  to  the  unit  mass,  and  cv  the 
speoific  heat  at  constant  volume,  expressed  in  work  units,  (8) 
may  be  written 

dQ=  cvdT  +  pdv.  (9) 

Again  it  is  plain  that 

dU  =  cvdT,  (10) 

2  1 


482  Thermodynamics. 

and,  if  cp  be  the  specific  heat  at  constant  pressure  expressed 
in  work  units,  that 

cp  =  cv  +  R.  (11) 


Examples. 

1.  Calculate  the  difference  between  the  two  specific  heats  of  air,  being  given 
that  a  cubic  metre  of  air  at  a  temperature  of  0°C.  and  under  a  pressure  of 
760  mm.  of  mercury,  whose  density  is  13-6,  weighs  1*2932  kilogrammes. 

Arts.  0-069. 

2.  For  any  gas  whose  density  referred  to  air  is  d,  show  that 

0-069 

3.  Determine  the  quantity  of  heat  which  must  be  imparted  to  a  gas  to  enable 
'  i    it  to  expand  at  a  constant  pressure  p\  from  the  volume  v\  to  the  volume  V2. 

9  c 

Am.  Q  =  -~pi(vz  -  v\). 
K 

i.  If  T  be  the  absolute  temperature  of  a  gas,  and  ,9"  the  portion  of  the 
energy  of  its  unit  mass  which  is  due  to  the  velocities  of  translation  of  its  mole- 
cules, show  that  9'=MT. 

Since  pv  =  f&'  (Ex.  18,  Art.  288),  this  result  follows  from  (6)  Art.  324. 

5.  Determine  the  mean  velocity  of  translation  of  a  molecule  of  air  which  is 
at  a  temperature  of  0°  C. 

Here  T  is  273,  and  the  mean  velocity  required  is  485  metres  per  second, 
nearly. 

6.  Show  that  the  mean  velocities  of  translation  of  the  molecules  of  different 
gases  when  at  the  same  temperature  are  inversely  proportional  to  the  square 
roots  of  the  densities  of  the  gases. 

7.  Determine  the  relation  between  the  total  kinetic  energy  £?  of  a  gas  and 
that  portion  Sf'  of  the  kinetic  energy  which  is  due  to  the  velocities  of  translation 
of  its  molecules. 

The  total  energy  U  of  a  unit  mass  of  a  gas  is  composed  of  the  kinetic  energy 
^/,  and  of  the  potential  energy  V,  which  again  is  the  sum  of  two  parts,  I  \ 
resulting  from  the  mutual  action  of  the  molecules,  and  Y%  depending  on  the 
constitution  of  the  individual  molecules.  V2  may  be  considered  constant  so 
long  as  the  chemical  constitution  of  the  gas  remains  unchanged,  and  V\  may  be 
assumed  to  be  zero,  since  Z7  is  a  function  of  the  temperature  alone,  and  Fi,  if  it 
existed,  would  depend  on  the  mutual  distances  of  the  molecules,  and  therefore 
on  the  volume.     Hence  U=£/+  V%. 

Again,  —  =  JCV  =  constant,  whence  U=JCVT+  C,  or  &  +  Vz  =  JC„T+  C". 

Let  &=&&',  then  &=  %0RT,  and  %$RT +  F2  =  JCVT+  C.    Hence,  as 

Vi  is  constant,  $  must  be  of  the  form  7  +  -^ ,  where  7  and  y'  are  constants ;  and 


Indicator  Diagram.  483 

fc7must  be  the  sum  of  two  parts — one  proportional  to  the  temperature,  the  other 
constant.  The  existence  of  the  latter  part  seems  in  the  highest  degree  improb- 
able :   we  may,  therefore,  conclude  that  £  is  constant.     To  determine  its  value 

1  C 

we  have  ZQJl  =  JCV,  whence  fi  =  §  - — -  by  (7),  where  k=  -£■ .     Now  Tc  is  found 

K   —    1  Cy 

to  be  almost  the  same  for  all  gases,  and  to  be  equal  to  1-408  ;  hence  £  is 
approximately  the  same  for  all  gases,  and  is  equal  to  1-634. 

8.  Two  masses  of  different  gases  have  equal  volumes  at  the  same  pressure 
and  temperature ;  show  that  for  all  equal  temperatures  they  have  equal  kinetic-, 
energies. 


325.  Reversibility  and  Cyclical  Processes. — When 
a  body  experiences  transformations  such  that  the  inverse 
chauges  can  take  place  in  precisely  the  same  circumstances, 
the  transformation  is  said  to  be  reversible.  In  order  that  this 
should  be  the  case,  any  source  from  which  the  body  derives 
heat,  or  to  which  the  body  imparts  heat,  must,  at  the  time  at 
which  the  heat  is  transferred,  be  of  the  same  temperature  as 
the  body ;  and  also  the  external  pressure  on  the  body  at  any 
time  must  be  equal  to  the  pressure  corresponding  to  the  state 
of  the  body  at  the  time. 

A  cyclical  process  is  a  transformation  at  the  end  of  which 
the  body  returns  to  the  same  state  as  that  in  which  it  was  at 
the  beginning. 

326.  Indicator  Diagram. — The  state  of  a  body  is,  as 
we  have  seen,  a  function  of  two  independent  variables.  If 
those  selected  be  the  volume  of  the  unit  of  mass  and  the 
pressure  on  the  unit  of  area,  the  state  of  the  body  at  any 
time  is  indicated  by  the  position  of  a  point  whose  coordi- 
nates referred  to  two  rectangular  axes  are  proportional  to 
the  volume  and  pressure. 

In  the  case  of  a  body  undergoing  a  transformation  accord- 
ing to  a  fixed  law,  the  set  of  points  indicating  its  successive 
states  form  a  curve.  In  a  reversible  transformation,  if  no 
heat  be  lost  or  gained  by  the  body  during  the  transformation, 
this  curve  is  called  an  adiabatic  or  isentropic  curve.  If  the 
temperature  remain  constant  the  curve  is  called  isothermal. 
The  area  comprised  between  the  curve,  its  extreme  ordinates, 
and  the  axis  of  abscissas,  represents  the  work  done  by  the 
body  during  the  transformation. 

2  12 


484  Thermodynamics. 

327.  Isothermal*  and  Adiabatics  for  a  Perfect 
Gas. — In  the  case  of  a  perfect  gas  the  isothermal  curve  is 
determined  by  the  equation 

pv  =  ETly  (12) 

where  Tx  is  the  constant  temperature.     The  isothermal  for  a 
perfect  gas  is  therefore  an  equilateral  hyperbola. 

Since  the  temperature  remains  constant  the  heat  required 
to  effect  the  transformation  is  given  by  making  dT  =  0  in  (8), 
that  is  by  the  equation 

jq  =  rpdv  =  MT1["-  =  BTl  log  f-2Y       (13) 

When  a  body  undergoes  an  adiabatic  transformation, 
dQ  =  0,  and  therefore  in  this  case  (8)  becomes 

JCvdT  +  pdv  =  0.  (14) 

If  we  substitute  in  this  the  values  of  dT  and  R  derived  from 
(6)  and  (7),  and  put  Cp  =  Wv,  we  get 

rln-i  dt) 

kpdv  +  vdp  =  0,     that  is     k  —  +  —  =  0. 
11  v       p 

Integrating  we  have 

pvk  =  PiVik,  (15) 

where  px  and  vx  are  the  initial  pressure  and  volume. 

The  temperature  T  at  any  stage  of  an  adiabatic  trans- 
formation is  given  by  the  equation 

T,     PM      \vj  v     } 


Examples. 

1.  In  an  adiabatic  transformation  determine  the  equation  connecting  th( 
initial  and  final  pressures  of  a  gas  with  its  initial  and  final  volumes. 


pi     w 


2. 
tion. 


Fundamental  Principles  of  Thermodynamics,  485 

Determine  the  external  work  done  by  a  gas  in  an  isothermal  transforma- 


V 


In  this  case  W=JQ  =  RTi  log  -  =  pm  log  -. 

3.  Prove  that  the  external  work  done  by  the  unit  mass  of  a  gas  in  an  adiabatic 
transformation  is  JCV(TX  -  T2),  where  T\  and  T%  are  the  initial  and  final  tem- 
peratures. 

4.  If  the  decrease  in  the  temperature  of  the  air  as  its  height  above  the 
surface  of  the  earth  increases  were  due  merely  to  the  fall  of  temperature  result- 
ing from  the  expansion  caused  by  diminution  of  pressure,  show  that  AT,  the 
excess  of  the  temperature  at  the  earth's  surface  above  the  temperature  at  any 
height  s,  would  be  given  by  the  equation 

£-1273* 

where  ho  is  the  height  of  a  homogeneous  atmosphere  at  0°  C. 

If  p  be  the  pressure,  and  p  the  density  of  the  air  at  the  height  z,  we  have, 

1        p 
from  the  fundamental  equation  of  hydrostatics,  dp  =  -  gpdz;  but  p  =  -  =  — -  , 

,.  ,     •      dT       k-1  dp       __..,. 
and  since  the  expansion  is  adiabatic,   —    =    — .      Eliminating  dp,  we 

have^T= - — "S^j  from  this,  since  gho=poVo,  we  obtain  the  equation 

k     It 

given  above. 

328.  Fundamental  Principles  of  Thermodyna- 
mics.— The  science  of  Thermodynamics  is  founded  on  two 
fundamental  Principles.  Of  these,  the  first  finds  its  mathe- 
matical expression  in  Equation  (1),  and  involves  two  state- 
ments, viz.  that  In  ever//  natural  process  the  total  energy  is 
invariable;  and  that  Seat  is  a  form  of  energy,  a  definite  amount 
of  heat  being  equivalent  to  a  definite  amount  of  work. 

The  second  fundamental  Principle  was  first  stated  by 
Clausius,  as  follows: — It  is  impossible  for  a  machine,  unaided 
by  external  energy,  to  convey  heat  from  one  body  to  another  at  a 
higher  temperature. 

By  Thomson  the  same  Principle  is  stated  somewhat  diffe- 
rently in  the  following  manner : — It  is  impossible  by  means  of 
inanimate  material  agency  to  derive  mechanical  effect  from  any 
portion  of  matter  by  cooling  it  beloiu  the  temperature  of  the 
coldest  of  the  surrounding  objects  ;  and  by  Clerk  Maxwell  in 
another  form,  thus  : — It  is  impossible,  by  the  unaided  action  of 
natural  processes,  to  transform  any  part  of  the  heat  of  a  body 


v* 


486 


Thermodynamics. 


C*D* 


into  mechanical  work,  except  by  allowing  heat  to  pass  from  that 
body  into  another  at  a  lower  temperature. 

This  Principle  merely  expresses  the  teaching  of  expe- 
rience in  reference  to  the  connexion  between  temperature 
and  the  transference  of  heat. 

329.  Carraot's  Cycle. — Let  us  suppose  a  body  subject 
to  a  reversible  cyclical  process  indicated  by  an  isothermal 
AXBX,  an  adiabatic  BXB2,  another  isothermal  B,A2,  at  a  lower 
temperature  than  the  former,  and 
another  adiabatic  A2AX.  Whilst 
expanding  from  the  volume  repre- 
sented by  OCx  to  that  represented 
by  ODx  the  body  is  kept  at  a  con- 
stant temperature  tXi  receiving  from 
a  source  Kx  of  heat  at  that  tempera- 
ture a  quantity  of  heat  Qx.  Whilst 
expanding  from  ODx  to  OD2,  the  q~ 
temperature  falls  from  tx  to  t2,  no 
heat  being  lost  or  gained.  The  body  is  now  compressed  from 
OD2  to  OC2,  and  the  heat  Q2  thereby  developed  is  imparted 
to  a  reservoir  K2  at  the  temperature  t2.  Finally,  the  body  is 
compressed  from  OC2  to  OCXi  and  the  temperature  thereby 
rises  from  t2  to  tx,  no  heat  being  lost  or  gained.  In  this 
process  the  volume  at  which  the  adiabatic  compression 
commences  is  selected  so  that  when  the  volume  returns  to 
its  initial  value  the  temperature  returns  likewise  to  its  initial 
value.  In  the  whole  process  Qx  units  of  heat  are  imparted  to 
the  body,  and  Q2  units  of  heat  are  taken  from  it ;  and  as  the 
body  returns  to  its  original  state,  Qx  -  Q2  units  of  heat  have 
been  transformed  into  the  work  which  is  indicated  by  the 
area  AXBXB2A2.  In  referring  to  this  process  Kx  and  K2  are 
frequently  termed  the  source  and  the  condenser. 

We  can  now  show  that,  if  Qx,  tx,  and  t2  be  supposed  in- 
variable, Q2  must  be  the  same  for  all  bodies. 

Suppose  Q2  for  one  body  M  were  greater  than  for  another 
body  i,  its  value  for  L  being  denoted  by  Q2,  and  for  M  by 
Q'2.  Employ  the  cyclical  process  for  the  body  L  to  work 
that  for  the  body  31  in  reverse  order.  This  can  be  done 
because  Qx  -  Q2  >  Qx  -  Q'2.  Then  the  source  of  heat  at  tx 
remains  as  before,  whilst  the  source  of  heat  at  t%  has  received 


Determination  of  Carnofs  Function.  487 

Q:  and  given  out  Q2  units  of  heat.  Moreover,  an  amount  of 
work  represented  by  Q2  -  Q2  has  been  accomplished.  The 
result  of  the  whole  process  is  that  work  has  been  done  by 
means  of  heat  obtained  from  the  coldest  body  in  the  system. 
As  this  result  is  opposed  to  the  second  fundamental  Principle 
(Art.  328),  we  conclude  that  Q2  is  the  same  for  all  bodies,  and 
is  therefore  simply  a  function  of  Qlt  tu  and  t2. 

From  what  has  been  now  proved  it  follows  that  if  we  sup- 
pose the  curve  AlBl  divided  into  n  parts,  for  each  of  which  Qx 
is  the  same,  and  adiabatics  drawn  through  the  points  of 
section,  the  corresponding  values  of  Q2  are  equal.  Hence  it  is 
easy  to  see  that  Q2  becomes  n  Q2  if  Qy  become  n  Qi,  and  therefore 

that  -=r  must  be  independent  of  Qx ;  accordingly,  we  have 
|  =/(*.,  tt).  (17) 

Again,  if  W  be  the  amount  of  heat  converted  into  work 
in  the  process,  we  get,  from  (17), 

J-l-/ft*<*  (18) 

330.    Determination    of    Carnot's    Function. — In 

order  to  determine  the  function/  we  have  merely  to  select  a 
body  for  which  the  isothermal  and  adiabatic  curves  are  known. 
Let  us  then  select  a  perfect  gas. 
In  this  case,  by  (13), 

JQ1  =  RTX  log  5»a,     JQ2  =  RT2  log  &. 

Again,  as  the  points  Ax  and  A2  lie  on  the  same  adiabatic,  by 
(16)  we  have 

fcf^.and  likewise  tef-£; 
therefore  —  =  — -,  and  7r  =  — . 


488 


Thermodynamics. 


Hence,   whatever   be   the   body   employed,   we   obtain   the 
equation 

ft     Q2 


T, 


(19) 


331.  Extension  of  Carnot's  Cycle. — If  heat  imparted 
to  a  body  be  regarded  as  positive,  and  heat  given  out  by  the 
body  as  negative,  (19)  may  be  written 


rA  +  T2 


■^  +  ^  =  0. 


(20) 


If  we  now  suppose  a  reversible  cyclical  process  represented 
by  any  number  of  isothermals  and  adiabatics,  each  isothermal 
being  followed  by  an  adiabatic,  and  if  Q  be  the  number  of 
units  of  heat  imparted  to  the  body  at  the  temperature  T, 

Q 

we  have  the  equation  2^=0. 

In  order  to  prove  this,  let  us  first 
suppose  a  cycle  in  which  there  are  three 
isothermals,  A^Bi,  B2C2,  and  AzCz,  cor- 
responding to  the  temperatures  Tx,  T2, 
and  Tz.  Produce  the  adiabatic  BXB2  to 
Bz,  then  Qz=qz  +  q*,  where  qz  corresponds 
to  BZA 3,  and  qz  to  CZBZ. 

Now  by  (20), 

Qi      ft      n  ,  Qz      q'     ft 

-+-  =  0,   axui-+-  =  0 


from  which,  by  addition,  we  have 

Qi     0*     Qz 

r, +  r8  +  tz 


o. 


This  result  "may  be  extended  in  a  similar  manner  to  a  cycle 
containing  four  isothermals,  and  so  on.     Hence,  in  general, 


4=o. 


(21) 


Entropy.  489 

Again  (21)  holds  good  for  every  reversible  cyclical  process, 
whatever  be  the  nature  of  the  curves  by  which  it  is  repre- 
sented. 

This  appears  from  the  consideration  that  two  infinitely 
near  points  A  and  B  on  any  curve  can  be  connected  by  the 
element  of  an  isothermal  followed  by  that  of  an  adiabatic,  and 
that  the  area  bounded  by  these  elements,  the  ordinates  of  A 
and  B,  and  the  axis  of  abscissas,  differs  only  by  an  infinitely 
small  quantity  of  the  second  order  from  the  area  of  which  the 
arc  AB  is  the  boundary. 

For  every  reversible  cyclical  process,  however  effected,  we 
have,  then,  the  equation 

jf  =  o.  (22) 

332.  Entropy. — If  a  body  pass  from  any  one  state  to 
any  other,  we  may  suppose  the  change  of  state  effected  by 
means  of  a  reversible  transformation ;  and,  whatever  this  pro- 
cess be,   -=■  between  the  limits  corresponding  to  the  two  states 

must  have  the  same  value,  since  the  cycle  may  be  completed 

[clQ 
by  a  definite  invariable  transformation.     Hence    —  depends 

only  on  the  state  of  the  body,  and  is  independent  of  the  mode 
(supposed  reversible)  by  which  the  body  is  brought  into  this 
state. 

If  we  put    —  =  </>,  the  quantity  <p  is  called  by  Clausius 

the  Entropy  of  the  body. 

The  second  fundamental  Principle  of  Thermodynamics 

leads  therefore  to  the  result,  that  the  entropy  <p  is  a  function 

of  the  two  independent  variables  on  which  the  state  of  the 

^body  depends,  and  therefore  that  in  all  reversible  trans/or- 

dQ 
/nations  —  is  a  perfect  differential  dcp,  or  that 

dQ  =  Td<p.  (23) 

In  theoretical  applications  of  the  equation  of  entropy, 
Q  and  <j>  are  supposed  to  be  expressed  in  mechanical  units. 


490  Thermodynamics. 

333.  Energy  and  Entropy. — For  every  reversible 
transformation  in  which  the  external  work  done  by  the  body 
is  due  to  its  own  expansion  we  have,  if  Q  be  expressed  in 
work  units,  the  two  equations 

dQ  =  dU  +  pdv) 

(24) 
dQ  =  Td<j>  ) 

The  energy  U  and  the  entropy  <£  are  functions  of  the 
independent  variables  on  which  the  state  of  the  body  depends, 
and  dU  and  dip  are  therefore  perfect  differentials ;  Q  depends 
not  merely  on  the  state  of  the  body  but  also  on  the  mode 
in  which  it  has  been  brought  into  that  state  ;  hence  dQ 
is  not  a  perfect  differential.     The  limits  of  the  quantities 

,  ,  &c.  are  expressible  in  terms  of  the  independent 

Av     Ap 

variables  and  the  differential  coefficients  of  JJ  and  v.    They 

are  therefore  functions  of  the  two  independent  variables  which 

determine  the  state  of  the  body,  but  are  not  differential  coeffl- 

£Q 
cients.     They  may  be  written  -^,  &c. 

cv 

Again  from  equations  (24),  we  have 

dU=  Td<p  -pdv.  (25) 

In  this  equation  if  we  select  successively  as  independent 
variables  <p,  v;  <f>,  p;  T,  p  ;  T,  v ;  and  v,  p  ;  and  express  in 
each  case  the  condition  that  dU  should  be  a  perfect  diffe- 
rential, we  obtain  a  system  of  equations  which  hold  good  in 
any  reversible  transformation  in  which  the  external  work 
done  by  a  body  is  due  to  its  expansion  against  the  pressure 
on  its  surface,  and  which  are  as  follows  : — 

dvh~     WA'     WA"WV     VpJt"     \dT)p\      .... 

d±\=(dp\       (d_T\  (dj\  _  (clT\  (d£\  m 

dv)T    \dT)J     \dpJv\dvjP    \dv)p\dp)v  J 


Elasticity  and  Expansion.  491 

Briot  remarks  that  from  the  first  of  these  equations  the 
three  succeeding  can  be  obtained  by  interchanging  p  and  i\ 
or  T  and  <p,  the  sign  of  the  right-hand  member  of  the  equa- 
tion being  altered  after  each  interchange. 

334.  Elasticity  and  Expansion. — The  elasticity  of  a 
substance  may  be  defined  as  the  limit  of  the  ratio  of  an  in- 
crease of  pressure  to  the  compression  which  it  produces,  the 
compression  being  the  ratio  of  the  diminution  of  volume  to 
the  original  volume. 

The  state  of  a  substance  being  determined  by  two  inde- 
pendent variables,  except  some  connexion  between  their  va- 
riations be  assigned,  the  elasticity  is  indeterminate.  The  two 
elasticities  usually  considered  are  the  elasticity  at  constant 
temperature  ET,  and  the  elasticity  at  constant  entropy  E^. 
The  former  obviously  belongs  to  an  isothermal,  and  the  latter 
to  an  adiabatic,  transformation. 

From  the  definitions  of  Er  and  E$  we  have 

*— @&  *— &■    w 

"When  a  body  is  heated  it  usually  expands.  The  expan- 
sion is  the  ratio  of  the  increase  of  volume  to  the  original 
volume,  and  the  expansibility  is  the  limit  of  the  ratio  of  the 
expansion  to  the  increase  of  temperature,  the  pressure  re- 
maining constant.     If  e  denote  the  expansibility,  we  have 


"H3L-  (28) 

If  the  expansion  of  a  body  take  place  without  change  of 
temperature,  the  limit  of  the  ratio  of  the  heat  required  for 
the  expansion  to  the  increase  of  volume  is  called  the  latent 
heat  of  expansion,  and  may  be  denoted  by  I;  hence 


J\\  dv  Jt 

The  expansibility  of  a  substance  is  called  by  some  writers 
its  coefficient  of  dilatation. 


492  Thermodynamics. 

Examples. 

1.  The  volume  and  pressure  of  a  gas  being  given,  determine  its  entropy. 

Ans.  <p  -  <p0  =  J  \Cv  log  —  +  Cp  log  —  [ . 
(  Po  Vo) 

2.  Show  that 

\dv  J  T 

The  first  two  of  these  equations  follow  at  once  from  (23) .     To  prove  the  last, 
we  have  from  the  two  former, 

{&).-l$b@),&k 

and  by  (26), 

Again,  dp  =  I  —  J    dT  +  I  —  j     dv,  from  which,  by  making  dp  =  0,  we  get 

f  —J  ,  and  substituting  in  the  expression  for  J {Cp  —  Cv)  already  given,  we 
obtain  the  result  required. 
Cp      I 


3.  Prove  that     °p  _^» . 


/d$\         (ty\    (dv\ 

cridT)=  m  m  >  but  d*  =  {d-v)Pdv+[dp)vd*> 

/#\  ldT\ 

and  therefore  1&L  =  -   { ±)   .     In  like  manner  iftil  =  -  (±)     ; 

\dp)v  \dv)p 

idp\ 

hence  £  =  A*l2±  =  f»  (Art.  334). 

\dv  J  t 


Examples.  493 

4.  Prove  that  dQ  =  cpdT  -  cvTdp. 

"- ®."*(S),* 

"*"+'(8)r* 

but  by  (26)  -we  have  (—  )    =—  (  t^,)  ,  and  hence  by  substitution  we  obtain 
from  (28)  the  required  result. 

5.  Assuming  that  the  square  of  the  velocity  of  the  propagation  of  sound  is 
proportional  to  the  elasticity  of  the  medium  divided  by  its  density,  show  that  in 
a  gas  the  velocity  of  sound  varies  as  \ZkBT. 

Since  the  compression  of  the  air  during  the  passage  of  a  wave  of  sound  is 
very  sudden,  the  compression  may  be  regarded  as  adiabatic.  Hence  the  velocity 
of  sound  varies  as  ^ E^v,  but  E$  —  JcET  (Ex.  3),  and  E  T=  P,  therefore,  &c. 

By  means  of  the  results  obtained  in  this  Example  and  in  Ex.  1,  Art.  324, 
if  the  velocity  of  sound  be  determined  by  experiment,  Cp  and  Cv  can  be  calculated. 
Conversely,  if  Cp  be  known  by  experiment,  Cv  can  be  found  from  the  velocity 
of  sound,  and  hence  the  value  of  /  can  be  determined. 

6.  Show  that  bodies  which  expand  by  heating  are  heated  by  compression  ; 
those  which  contract  by  heating  are  cooled  by  compression ;  and,  if  the  tempera- 
ture be  maintained  constant,  determine  the  rate  at  which  heat  is  given  out  or 
absorbed  according  as  the  pressure  is  increased. 

If  Q  be  the  heat  required  to  keep  the  temperature  constant,  the  rate  of  ab- 
sorption is  (  —  J     ;  but 

(S)'-*(2),--*S),-:--E  *-«-»>■ 

Hence  8Q  is  negative  if  e  be  positive,  and  conversely. 

7.  Prove  that  in  water  not  far  from  its  maximum  density  the  rise  of  tem- 
perature produced  by  an  increase  of  pressure  is  given  approximately  by  the 
formula, 

2950000  *' 

where  t  is  expressed  in  degrees  centigrade,  and  p  in  atmospheres. 

If  vo  be  the  volume  of  the  unit  mass  of  water  at  4°,  when  the  density  is  a 

maximum,  the  empirical  formula  v  =  vq  f  1  +  )    represents,  according 

to  Kopp  and  Tait,  the  results  of  numerous  experiments.     From  this  formula  we 

have  approximately  e  =  --^qqq- 

Hence,  assuming  the  pressure  of  the  atmosphere  to  be  1033  grammes  on  the 
square  centimetre,  we  obtain  the  required  result. 

8.  If  the  internal  energy  of  a  body  be  a  function  of  its  temperature  alone 
determine  the  relation  which  must  exist  between  v,  p,  and  T. 


494  Thermodynamics. 

In  this  case  (25)  becomes  Td<p  =  —  dT  +  pdv,  whence 

dU  dT      p  J 
d*-dT-T  =  TdV' 

The  left-hand  side  of  this  equation  is  a  perfect  differential,  and  therefore 
p  =  Tf(v),  which  is  the  relation  required. 

9.  If  a  body  be  such  that  its  energy  increases  uniformly  with  the  tempe- 
rature when  the  volume  is  constant,  and  uniformly  with  the  volume  when  the 
temperature  is  constant,  and  that  its  specific  heat  at  constant  pressure  is  con- 
stant, determine  the  equation  connecting  volume,  pressure,  and  temperature. 

Here  we  must  have  dU=  adT -f  bdv,  where  a  and  b  are  constants.  Hence 
from  (25)  we  get  Td<f>  =  adT  +  (b  +  p)  dv,  whence  b  +  p  =  Tf(v).  If  by  means 
of  this  last  equation  we  express  dv  in  terms  of  dp  and  dT,  we  have 


dQ 


=  {a-7)iT+h-  Now*-Gf), 


P 

and  therefore,  if  n  be  the  constant  value  of  cp,  we  obtain  a-'—  =  n.     From  this 

we  have  [a  -  n)  ■£■  =  dv,  and  integrating  we  get  (v  +  C)  f=  (n  -  a),  where  0 
is  the  constant  of  integration.     Hence  we  have  as  the  required  relation 

(b+p){v+  C)  =  {n-a)T. 

10.  If  the  specific  heats  of  a  body  at  constant  pressure  and  at  constant 
volume  be  each  constant,  show  that  the  energy  is  a  linear  function  of  the  volume 
and  absolute  temperature. 

~Letcv  =  m,     cP  =  n,  then  Ijj,)     =  %   and  therefore    U  =  mT  +  f  (v). 

Ah0C*={§)P+p  (is),- whence "  -  •" + (/' +p)  (%)„■    (a) 

Again,  from  (25)  we  have 

Td<p  =  widT  +  (/'  +  p)  dv,     whence    f'  +  p=  TF{v).  (b) 

If  we  differentiate  this  equation,  and  eliminate  from  the  equation  so  obtained 

and  equations  (a)  and  (b)  the  two  quantities  p  and  (  — )   ,  we  get 

\dTj  p 

{n  -  m)  f"  =  T{F*+  (n  -  m)  F'} . 

This  equation  cannot  be  true  for  all  values  of  the  independent  variables 
T  and  v  except  each  side  vanish  separately,  hence  we  have  /"  =  0,  and 
therefore /(v)  =  C\v  +  Co.     Consequently  XT  =  mT+  C\v  +  C2. 


Non-reversible  Transformations.  495 

11.  If  the  speciBc  heat  of  a  body  at  constant  volume  be  constant,  and  the 
expansibility  at  any  temperature  be  the  inverse  of  the  absolute  temperature, 
determine  the  equation  connecting  volume,  pressure,  and  temperature,  and  find 
the  energy  in  terms  of  the  temperature  and  volume. 

Here,  as  in  Ex.  10,  U=mT+f{v),andf(v)+p  =  TF(v).   Also  -l—\  =  i 

whence  v  =  T\\i{p).     Eliminating  fwe  have  vF[v)  =  ty(p){f'(v)  +  p). 
Differentiating  with  respect  to  p  we  get 

This  equation  cannot  be  true  in  general  except 

f  (v)  =  -  C,  and  }{p)  +  pf(p)  =  CV(p), 

where  Cis  constant. 

Hence  we  obtain  {C - p)  v  =  XT,  and  U  =  mT  -  Cv  +  C\  where  K  and  C 
are  constants. 

335.  ^on-reversible  Transforinations. — In  the  case 
of  a  non-reversible  transformation  we  cannot  assume  the 
truth  of  equations  (24).  In  fact,  for  such  a  transformation, 
even  through  the  external  work  done  by  the  body  be  due  to 
its  expansion  against  external  pressure,  this  pressure  need  not 
be  equal  in  magnitude  to  that  belonging  to  the  state  of  the 
body,  nor  is  dQ  in  such  a  transformation  necessarily  equal  to 
Tty. 

In  this  case  we  must  proceed  as  follows  : — Let  H  be  the 
heat  actually  imparted  to  the  body  in  the  non- reversible 
process,  W  the  external  work  done,  and  Un  and  U  the  initial 
and  final  energies  of  the  body  ;  then 

E=  U-  U0+  W.  (30) 

Let  us  now  imagine  a  reversible  transformation  capable  of 
bringing  the  body  from  its  initial  to  its  final  state,  but  other- 
wise perfectly  arbitrary.  Since  this  hypothetical  transforma- 
tion is  reversible,  we  can  make  use  of  equations  (24)  and  (25), 
and  of  any  results  therefrom  deducible  to  assist  in  determining 
U  -  U0.  The  expression  thus  obtained  may  be  substituted 
in  (30). 


496  Thermodynamics. 


Examples. 

1 .  A  gas  at  p\  and  V\  is  allowed  to  expand  into  a  perfectly  empty  vessel, 
whereby  its  pressure  and  volume  become  p%  and  v%.  No  heat  being  imparted 
to  the  gas  or  taken  from  it,  determine  its  change  of  temperature. 

Since  no  external  work  is  done,  and  no  heat  lost  or  gained,  the  energy  must 
remain  constant,  and  therefore  the  temperature  (Art.  324).  The  assumption 
that  the  energy  of  a  gas  is  a  function  of  the  temperature  alone,  is  indeed  a  result 
from  the  fact  ascertained  by  experiment,  that  the  temperature  is  constant  under 
the  circumstances  here  supposed. 


2.  In  the  preceding  example  determine  the  change  of  entropy. 
We  have  (Ex.  1,  Art,  334), 

<*>,  _  ^  =  /  ( Cv\og-+  Cp  log  -)  =J(CP-  Cv)  log  -  =  R  log  -. 


Since  vz  >v\,  the  entropy  is  increased  by  the  supposed  transformation.  This 
transformation,  it  should  be  observed,  is  non-reversible,  and  therefore  not  adia- 
batic,  though  no  heat  is  lost  or  gained. ' 

3.  A  vertical  cylinder,  whose  horizontal  section  is  S,  is  filled  with  gas  at  the 
atmospheric  pressure  p\  and  temperature  1\,  and  closed  by  a  piston  on  which 
is  placed  a  weight  w  which  pushes  it  down.  Supposing  no  external  heatto 
pass  into  or  out  of  the  gas,  determine  the  temperature  when  equilibrium  is 
established. 

The  transformation  here  is  non- reversible,  since  the  external  pressure  ex- 
ceeds that  due  to  the  state  of  the  body  by  a  finite  amount.  Since  no  heat  is  lost 
or  gained  the  external  work  done  on  the  gas  must  be  equal  to  the  change  of 
energy. 

Let  pz,   Ti,  V2  be  the  final  pressure,  temperature,  and  specific  volume,  the 

w 
initial  specific  volume  being  vi,  then  pi  =  p\  +  -,  and  the  work  done  on  the  unit 

of  mass  is  pz  {v\  -  vz).    Hence  from  (10)  we  have 

c„  (r2  -  Ti)  =p2  {vi  -  vz)  =  P2V1  -  R  T2, 


and  therefore 


{fin  +  R)  T2  =  cv  Tx  4   (px  +  ^\  vi, 


4.  In  Ex.  3  determine  the  increase  of  the  entropy  of  a  unit  mass  of  the  gas. 

Tz       „       ,.  ,      vi) 


Tz  =  cp  T\  +  —  v\,  which  determines  Tz 
increase  of  the 
Am.  <pz-  <p\  =  cv  jlog  -f  —  {k  -  1)  lo£ 

ori 


5.  In  Ex.  3  if  the  gas  were  adiabatically  compressed  from  its  original  to 
its  final  volume  determine  the  final  temperature  Tz'. 

Am.     Tz  =  Tx 


6.  Show  that  Tz  in  Ex.  3  is  greater  than  Tz  in  Ex.  5. 


Absolute  Scale  of  Temperature.  497 

The  equation  which  determines  T2  may  be  written  in  the  form 
*  T2  =  *,  ft  +  —  (vx  -  v9). 

V2 

If  we  put  v\  =  av2,  and,  as  usual,  cp  =  heVi  this  becomes 
Tz{l-(k-l)(a-l)}  =  T1. 

Let  k  -  1  =  u,     a  -  1>  $,     then  T2  =  Ti  (I  -  ufl)-\ 

also,  B*  5,    T2'=Tiav  =  T,  jl  +  *  log  (1  +  jB)  +  ^  log^l  +  0)  +  &c. 

but  j8  >  log  (1  +  j8),  for  e*3  >  1  +  0  =  el0§(1+|S>. 

Hence  each  term  of  the  series  for  T2  is  greater  than  the  corresponding  term  of 
the  series  for  T2,  and  as  all  the  terms  are  positive,  T2  >  T/. 

The  increase  of  entropy  in  the  gas  compressed,  as  in  Ex.  3,   might  be 
expressed  in  the  form 


<po  -  <px 


-**(S) 


7.*Two  vessels  impervious  to  heat,  whose  volumes  are  V\  and  V2,  contain 
gas  at  pressures  p\  and  p2,  and  at  the  same  temperature  T.  The  vessels  are 
placed  in  communication  with  each  other ;  determine  the  final  gain  of  entropy 
by  the  entire  system. 

Ans.  , ,  ^  =  Ym log  ffi  +  v*^  +  gg  iog  (ft  +  ^ 

8.  In  Ex.  7  prove  that  <p  —  <po  is  positive. 
Suppose  that  pi>  p2,  and  let 

1  +  = —  =  a,       1 =  va, 

Vipi  p\ 

then 

<P~  <!>o  =  -~1  {alog(l-y)-log(l-yo)}; 

and  since  a  >  1 ,  we  readily  see  that  the  quantity  inside  the  bracket  is  positive. 

336.  Absolute  Scale  of  Temperature. — The  result 
obtained  in  Art.  330  may  be  arrived  at  by  a  different  method 
independent  of  the  properties  of  any  particular  substance. 
We  have  seen  in  Art.  329  that  if  Q  be  the  heat  drawn  from 

2  K 


498  Thermodynamics. 

the  source,  and  W  the  heat  converted  into  work  in  Carnot's 

cycle,  —  is  a  function  of  the  extreme  temperatures  only,  and 

is  independent  of  the  suhstance  employed.  In  order,  then,  to 
construct  a  scale  of  temperature  independent  of  any  parti- 
cular body  we  may  proceed  as  follows  : — 

Draw  the  isothermal  AB  of  a  sub- 
stance chosen  at  random,  corresponding 
to  any  arbitrary  temperature,  which  may 
be  indicated  by  T,  and  draw  the  adiabatics 
AA  and  BR  corresponding  to  the  con- 
dition of  the  body  before  and  after  a 
certain  arbitrary  amount  of  heat  Q  has 
been  imparted  to  it. 

Draw  another  isothermal  at  a  tem- 
perature T'  less  than  T,  so  that  the  area 
ABB'  A'  may  be  of  given  magnitude  or 
correspond  to  a  given  amount  of  heat  w.     Now  draw  a  series 
of  isothermals  T" ,  T" ',  &c,  at  intervals  such  that 

ABBA  =  AB'B"  A'  =  A'B"B!"A"  =  &c. ; 

then  \IT-T  be  the  unit  of  temperature,  T-  T"  is  two  units, 
T-  T"  three  units,  &c. 

Since  T,  Q,  and  w  are  fixed  quantities,  and  W  correspond- 
ing to  T^  is  nw,  Equation  (18)  shows  that  two  bodies  are  at 
the  same  temperature  if  each  indicates  in  the  manner  described 
n  degrees  of  temperature  below  T.  This  method  of  estimat- 
ing temperature  is,  therefore,  independent  of  the  body  em- 

Again,  if  T  be  any  temperature  lower  than  T  estimated  in 
this  manner,  and  W  the  heat  converted  into  work  in  the  cor- 
responding cyclical  process,  we  have  W  =  (T-  T')  w,  and  in 
like  manner  for  another  temperature  T"  lower  than  T  we 
have  W"={T-T")w. 

If  we  now  suppose  a  cyclical  process  between  the  tem- 
peratures T  and  T",  indicated  by  the  points  A',  B%  B" ,  A\ 
the  heat  converted  into  work  is  W"  -  W\  and  we  get 

JT»-  W'={T'-T")  w  (31) 


Absolute  Zero.  499 

Again,  the  heat  Q/  drawn  from  the  source  at  Tf ,  is  equal 
to  that  given  to  the  condenser  in  the  process  in  which  T  and 
T'  are  the  extreme  temperatures  ;  hence 

Q  .  q  =  W'  =  (T-  T)  w,  that  is,  Q'  =  Q  -  (T-  T')  w.    (32) 

337.  Efficiency  of  a  Heat  Engine. — A  system  work- 
ing in  the  manner  required  by  Camot's  cycle  may  be  termed 
a  reversible  heat  engine,  and  the  ratio  of  the  heat  converted 
into  work  to  the  heat  drawn  from  the  source  is  called  the 
efficiency  of  the  engine. 

It  appears  by  the  reasoning  of  Art.  329  that  the  extreme 
temperatures  being  given,  the  efficiency  of  a  non-reversible  engine 
cannot  exceed  that  of  a  reversible,  and  that  the  efficiency  of  all 
reversible  engines  is  the  same. 

338.  Absolute  Zero. — From  Art.  337  it  appears  that 
the  efficiency  of   a  reversible  engine  working  between  the 

temperatures  T  and  T"  is j-f .    By  (31)  and  (32)  this 

H 

rpr  _   /xt// 

becomes  -. -^  • 

r\T-l) 

As  T"  decreases,  the  efficiency  increases,  but  the  limit 
which  it  can  never  exceed  is  unity,  since  the  mechanical  work 
done  by  an  engine  can  never  exceed  the  equivalent  of  the 
heat  drawn  from  the  source.  Hence,  if  we  make  the  effi- 
ciency unity,  we  obtain  for  T"  the  smallest  possible  value, 

which  is  T This  temperature  T" ,  since  it  is  the  lowest 

which  can  be  attained  by  any  body,  must  be  the  absolute 
zero.     Hence 

'-!-"•  - 14 

The  expression  for  the  efficiency  of  a  reversible  engine 
working  between  any  two  temperatures  Tr  and  T"  becomes 

2  K  2 


500  Thermodynamics. 

rpr  _  rprr 

then — ^7 — ,  and  for  the  cyclical  process  described  in  Art. 

Q  -  Qo      T  —To 

329  we  have  — -jz — -  =  — ^= — -.     Carnot's  function  has  thus 
H\  J-  i 

been  determined  independently  of  the  properties  of  any 
particular  substance. 

Again,  this  mode  of  determining  Carnot's  function  shows 
that  the  existence  of  an  absolute  zero  of  temperature,  sug- 
gested and  rendered  probable  by  the  known  properties  of 
what  are  called  permanent  gases,  follows  necessarily  from 
the  two  fundamental  Principles  of  Thermodynamics. 

The  experiments  of  Joule  and  Thomson  have  shown  that 
the  absolute  zero  is  273*7  below  zero  on  the  Centigrade 
scale,  or  460*66  below  zero  on  the  Fahrenheit.  This  is  very 
nearly  the  same  result  as  that  of  Article  324. 

Examples. 

1.  The  entropy  <p  being  defined  by  tbe  equation  dQ  =f{t)  d<p,  prove  that  in 
a  body  subject  to  the  equation  pv  =  Rf(t),  where  R  is  constant,  the  energy  is 
a  function  of  the  temperature  alone. 

Let/(0  =  T,  then  from  (2)  we  have  d<f>  =  — -  -f  B  — .     Hence  -dU  must 

be  a  perfect  differential,  whence  U=  F(T). 

2.  Gas  is  made  to  pass  uniformly  through  a  tube  in  which  a  porous  plug, 
such  as  cotton-wool,  is  placed.  No  heat  is  permitted  to  leave  the  gas  or  enter  it 
from  any  external  source  ;  determine  the  connexion  between  the  variations  of 
pressure  and  temperature  caused  by  the  plug. 

Since  the  density  of  the  gas  at  any  particular  cross  section  of  the  tube  does 
not  vary  during  the  experiment,  equal  masses  of  gas  pass  through  each  section 
in  the  same  time,  or  the  velocity  of  the  unit  of  mass  is  constant.  Again,  any 
energy  which  is  lost  by  frictien  is  restored  as  heat.  We  are  therefore  entitled 
to  assume  that  any  change  in  the  energy  of  the  gas  as  it  passes  through  diffe- 
rent parts  of  the  tube  is  due  to  the  work  done  on  it  or  to  the  work  which  it 

Suppose  two  cross  sections  A  and  B  of  the  tube,  one  on  each  side  of  the 
plug,  the  pressures  at  which  are  p\  and  p2.  As  a  small  quantity  dm  of  gas 
passes  A  the  pressure  driving  it  forward  does  work  on  it  whose  amount  is 
pividm.  At  the  same  time  dm  does  work  on  the  next  layer  of  gas  which  is 
equal  to  the  work  done  on  dm  when  passing  the  section  consecutive  to  A.  Thus, 
in  going  from  A  to  B  tbe  work  done  by  dm  and  the  work  done  on  dm  compen- 
sate each  other,  with  the  exception  of  pwidm  done  on  dm,  and  p2v2dm  done  by 
dm.  In  other  words,  in  the  passage  from  A  to  B  the  whole  external  work  done 
by  dm  is  (p2v2  -  pwi)  dm,  and  therefore,  since  no  heat  is  lost  or  gained,  we  have 

U2-  U\  +  P2V2.  -  p\V\  =  0. 

Now     U=jTd(j>-  J  pdv  =$Td<p-pv+$  vdp,  or  U+  pv  =  J  Td<p  +  j  vdp. 


Change  of  State.  501 

Hence  the  hypothetical  reversible  transformation  (Art.  335)  must  be  such  that 

K*&).«*M8),+-M-fc 

Substituting  cp  for  t(j£)    (Ex.  2,  Art.  334),  and-  (^)   for  (^j     by  (26), 

we  have  \\cpdT  +  (v  -  T  i-^\    J  dpi  =  0. 

Hence,  if  we  can  integrate  the  expression  under  the  integral  sign  in  this 
equation,   the  relation  between    Tz  -  T\  and  p%  -  p\  is  determined.     If  the 

gas  were  theoretically  perfect  (Art.  324),  we  should  have  T  ( -— J    =  v,  and  T% 

would  be  equal  to  T\.     This  is  found  not  to  be  the  case.     We  may  therefore 
conclude  that  no  gas  is  theoretically  perfect,  and  we  cannot  assume  either 

that  a  is  constant  or  that  T=  t  +  -. 

a 
From  the  equation  vp  =  voPo  (1  +  at),  if  we  consider  a  as  variable,  we  have 

We  may  assume  that  a  ( T- 1)  differs  from  a  constant  by  a  very  small  quanticy, 
and  likewise  that  —  is  very  small,  and  may  integrate  between  T\  and  T%  neglect- 
ing these  quantities,  since  T\  -  Ti  is  observed  to  be  small.  For  similar  reasons 
we  may  assume  that  [  —J  =  [  —  1  ,  and  that  cr  is  constant.  Hence,  inte- 
grating, we  have 

-  cp  {To  -  Ti)  =  2)oV0{l  +  at-  aT)\LOgp2  -  logpi) ; 


and  therefore 


1  cJTx-m 

T=t+-+ 


a      ap0v0  (log  pi  -  log  p2) 


From  this  equation  the  exact  position  on  the  centigrade  scale  of  the  absolute 
zero  can  be  determined  (Art.  338). 

339.  Change  of  State. — There  are  three  states  or  con- 
ditions in  one  of  which  matter  is  usually  found  ;  and  which 
are  termed  the  solid,  the  liquid,  and  the  gaseous. 

A  solid  body  is  one  which  strongly  resists  any  forces 
tending  to  alter  the  relative  positions  of  its  adjacent  mole- 
cules, and,  so  long  as  its  structure  is  unbroken,  admits  of  only 
slight  changes  in  these  positions. 


502  Thermodynamics. 

A  liquid  offers  scarcely  any  resistance  to  a  change  in  the 
mutual  position  of  its  molecules,  provided  this  change  does 
not  diminish  the  distance  between  those  which  are  adjacent. 
In  other  words,  it  is  indifferent  to  change  of  shape  or  sepa- 
ration of  its  molecules  from  each  other,  but  strongly  resists 
compression. 

A  gas,  like  a  liquid,  is  indifferent  to  change  of  shape,  but 
yields  to  compression  with  comparative  facility,  and  tends  to 
increase  its  volume  without  limit.  To  prevent  its  escape,  it 
must  therefore  be  restrained  by  an  external  envelope. 

It  is  almost  certain  that  every  substance  in  nature  is 
capable  of  existing  in  any  one  of  the  three  states,  and  passes 
at  a  certain  pressure  and  temperature  from  one  of  these  states 
into  another. 

Thus,  in  general,  there  is  a  certain  pressure  and  tempe- 
rature, at  which,  if  heat  be  imparted  to  a  solid  body  and  the 
pressure  be  maintained  constant,  the  temperature  does  not 
rise,  but  the  body  gradually  passes  into  the  liquid  state.  The 
amount  of  heat  required  to  bring  about  this  change  for  the 
unit  of  mass  is  called  the  latent  heat  of  liquidity.  If  the 
volume  of  the  liquid  exceed  that  of  the  solid,  the  latent  heat 
of  liquidity  is  spent  partly  in  altering  the  internal  energy  and 
partly  in  doing  external  work.  If  (as  is  the  case  with  water) 
the  volume  of  the  liquid  be  less  than  that  of  the  solid,  the 
internal  energy  of  the  liquid  exceeds  that  of  the  solid  by  an 
amount  greater  than  the  equivalent  of  the  latent  heat  of 
liquidity. 

The  vapour  of  a  liquid  may  be  considered  a  gas.  If  the 
temperature  be  sufficiently  high,  and  the  pressure  sufficiently 
low,  a  vapour  obeys  the  same  laws  as  the  gases  which  are 
called  permanent,  and  approaches  closely  to  the  condition  of 
a  perfect  gas  (Art.  324). 

For  each  vapour  there  is,  corresponding  to  any  given 
temperature  Tly  a  certain  pressure  pl9  such  that  at  higher 
pressures  the  vapour  begins  to  liquify  ;  and,  conversely, 
corresponding  to  any  given  pressure^  there  is  a  temperature 
Tiy  such  that  at  lower  temperatures  the  same  result  takes 
place.  The  pressure  px  is  called  the  maximum  pressure  of  the 
vapour  for  the  temperature  Tx,  and  Tx  is  called  the  boiling- 
point  of  the  liquid  for  the  pressure^. 


Change  of  State.  503 

In  fact,  if  heat  be  imparted  to  the  liquid  under  the  con- 
stant pressure  7^!,  the  temperature  of  the  liquid  will  rise  until 
it  reaches  Tx ;  after  this  the  temperature  will  remain  stationary, 
but  the  liquid  will  be  transformed  into  vapour,  and  the  heat 
required  by  the  unit  mass  for  this  transformation  is  called 
the  latent  heat  of  vaporization.  When  a  vapour  is  at  its 
maximum  pressure,  and  therefore  beginning  to  liquify,  it  is 
said  to  be  saturated. 

A  liquid  exposed  to  the  air  evaporates  more  or  less  at  all 
temperatures.  It  is  known  that  if  two  gases  be  enclosed  in 
the  same  envelope,  each,  after  some  time,  is  diffused  through 
the  whole  volume  of  the  envelope  as  if  the  other  were  absent, 
Hence  we  might  anticipate  the  behaviour  of  a  liquid  exposed 
to  the  atmosphere,  and  expect  that  the  air  would  not  act  like 
an  impervious  envelope  exercising  a  constant  pressure,  but 
would  merely  retard  the  formation  of  vapour  by  diminishing 
its  rate  of  diffusion. 

The  statements  of  Article  322  require  some  modification 
when  applied  to  a  body  while  changing  its  state.  So  long  as 
the  state  remains  unaltered,  Z7,  $,  and  v  are  functions  of  p 
and  T;  but  when  the  body  begins  to  change  its  state,  U,  0, 
and  v  vary,  even  though  p  and  T  remain  constant.  Whilst 
the  body  is  changing  its  state,  if  p  and  Tbe  constant,  U,  <p, 
and  v  are  functions  of  a  single  variable.  If  the  change  of 
state  go  on  continuously,  and  p  at  the  same  time  vary,  then 
T  must  also  vary,  and  be  at  each  instant  a  function  of  p. 
In  this  case,  TJ  and  <p  are  functions  of  two  independent 
variables,  which  may  be  v  and  p,  or  v  and  T,  but  cannot  be 
p  and  T. 

The  experiments  of  Andrews,  and  the  investigations  of 
Thomson,  have  thrown  much  light  on  the  phenomena  of 
change  of  state,  and  enable  us  to  explain  their  seeming 
anomalies.  An  account  of  these  researches  would,  however, 
be  outside  the  limits  of  the  present  work. 


504  Thermodynamics. 


Examples. 

I.  If  A  be  the  latent  heat  of  change  of  state  expressed  in  work  units,  and  v 
the  increase  in  the  volume  of  the  unit  mass  of  the  substance  after  passing  from 
one  state  to  the  other,  prove  that 

x  =  vTJt' 

Let  s  and  cr  be  the  volumes  of  the  unit  mass  of  the  substance  before  and 
after  the  change  of  state,  and  fi  the  fraction  of  the  unit  mass  which  has  undergone 
the  change  at  any  instant  during  the  transformation,  then  v  =  <r  —  s,  and 
v  =  fi<r  +  (1  —  n)s  =  s  +  vjx.  Again,  if  Q  be  the  quantity  of  heat  required  to 
transform  /j.  times  the  unit  mass  from  one  state  to  the  other,  the  pressure  and 
temperature  remaining  constant,  Q  =  A/t,  and  therefore 


S),-*®,— »),—&* 


(26). 


The  student  will  observe  that  p  is  here  a  function  of  T  alone,  and  that  if  L 
be  the  latent  heat  expressed  in  heat  units,  A  =  JL. 

2.  The  density  of  ice  being  0-92,  and  the  latent  heat  of  water  79-25,  find  the 
lowering  of  the  temperature  of  freezing  caused  by  an  additional  pressure  of  one 
atmosphere.  Ans.  0-0073°  C. 

3.  If  cs  and  cw  be  the  specific  heats  expressed  in  work  units  of  saturated 
steam  and  of  boiling  water  at  the  same  pressure,  show  that 

_  d\       A 
C,~Cw"dT~T' 

It  is  here  supposed  that  the  variations  of  Tand  p  are  so  related  (Art.  339) 
that  as  T  changes  the  steam  remains  saturated,  and  the  water  remains  boiling. 
Hence,  if  we  suppose  fx.  to  remain  constant  (Ex.  1),  we  have 

and  therefore 

hence,  substituting  and  performingthe  differentiation,  we  have  theresult  required. 
It  may  be  observed  that  cw  does  not  sensibly  differ  from  the  specific  heat  of  water 
at  constant  pressure. 

4.  Investigate  a  numerical  formula  for  the  specific  heat  of  saturated  steam 
at  any  given  temperature. 


Examples.  505 

According  to  Regnault,  the  whole  quantity  of  heat  required  to  raise  the  tem- 
perature of  the  unit  mass  of  water  from  0°  to  t°C.  and  evaporate  it  at  that 
temperature  is  606'5  +  0-305 1.     Hence  we  have  the  empirical  formula 

Z+f    Cwdt  =  606-5  +  0-305*, 


Jo 


where  Cw  is  the  specific  heat  of  hoiling  water  expressed  in  heat  units.     Diffe- 

dL 
rentiating,  we  have  —  +  Cw  =  0*305.      For  high   temperatures   Regnault's 

empirical  formula  may  he  replaced  hy  the  simpler  formula  of  Clausius,  viz. 

L  —  607  -  0-708 1.     If  we  express  — ,  by  means  of  the  latter  we  have  (Ex.  3), 


g.~0-30S-607-°-708'=  1-013      8°°-3 


273  +  t  273  +  * 

For  temperatures  near  0°  C.  we  may  take  Cw  =  1 ;  we  thus  get 

796-2 


C.=  l 


273  +  * 


From  the  expressions  obtained  for  Cs,  we  may  conclude  that,  except  the  tem- 
perature be  enormously  high,  the  specific  heat  of  saturated  steam  is  negative. 
Hence  it  follows,  that  if  saturated  steam  be  compressed,  the  temperature  after 
compression  will  be  higher  than  that  corresponding  to  saturation  at  the  new 
pressure  ;  or,  in  other  words,  saturated  steam  suffers  no  condensation,  but  becomes 
super-heated  by  adiabatic  compression.  Conversely,  if  saturated  steam  be  con- 
tained in  a  vessel  impervious  to  heat,  a  diminution  of  pressure  will  cause  partial 
condensation.  These  results  were  first  obtained  theoretically  by  Clausius  and 
Rankine,  who,  independently,  arrived  at  them  almost  simultaneously.  They 
have  since  been  confirmed  experimentally  by  Hirn.  It  seems  not  unlikely  that 
the  connexion  between  rain  and  a  change  of  atmospheric  pressure  depends  partly 
on  the  property  of  steam  mentioned  above. 

5.  If  U\  be  the  energy  of  the  unit  mass  of  saturated  steam  at  T\,  and  lh 
that  of  the  unit  mass  of  boiling  water  at  To,  prove  that 


oi.  *+£(«. -,£)«■ 


+  Ai  —  pwi. 


Let  us  suppose  that  the  unit  mass  is  brought  from  the  state  of  boiling  water 
at  To  and  p0  to  that  of  saturated  steam  at  T\  and p\,  and  that  this  transformation 
is  effected  by  first  bringing  the  water  without  evaporation,  but  continually 
boiling,  from  p0,  T0,  to ph  T\ ;  and  then  evaporating  at  T\,  p\. 

Take  as  variables  Tand  fx  (Ex.  1) ;  then,  since  dU '  =  dQ  —pdu,  we  have 

H®r'G)J-H(B.-'(S),«-- 


506  Thermodynamics. 

Again,  on  the  above  hypothesis,  when  Ovaries  p  is  zero,  and  when  p.  varies 
Tis  constant.     In  general  (Ex.  3), 

and  therefore  when  fx  =  0,  ( —J     =  c„,. 
\ol/  n 

Again, 

fdv\        ds        /5Q\  (dv\  ,_     ,, 

Hence,  substituting,  we  have 

Ui  =  U0+[   l(cw-p--^\dT+ (\i-2Jivi)\   dp. 

6.  Saturated  steam  in  a  vessel  containing  no  water  is  allowed  to  escape  into 
the  air ;  determine  the  quantity  of  heat  which  must  be  imparted  to  the  unit  of 
mass  in  order  that  it  should  remain  saturated. 

Let  Whe  the  external  work  done  by  the  unit  mass  of  the  steam  in  escaping, 
T\  and  T2  its  initial  and  final  temperatures ;  then  H  being  the  heat  required, 
we  have 

JH=  U2-  Ui+  W. 

NOW  W  =  p2{<T2  -  ffi)  =  p2{v2  -  Vi)  +  p2(s2  -  si),      (Ex.  1), 

where  a-  and  *  are  the  volumes  of  the  unit  mass  of  steam  and  of  water, 

f  Ta  (  ds  \ 

and  Uo-  Ui=\      k  -p  —  )dT+\z  ~  Al  -Wn+Pivi,  (Ex.  5). 

CT    I  ds  \ 

Hence  JH  =     2  lel0  -p  —  J  dT  +  A2  -  M  +  vi(pi  -  p2)  +  p%{s2  -  s\). 

Since  (s2  —  «i)  and  — —  are  small,  we  may  neglect  them,  and  thus  obtain 

JE=  1   %(clv^~\dT+vi{2Ji-p2). 

Now  \[cw  +^)=  0-305,     (Ex.  4)  ; 

J  \         dT) 

hence  H=  - 0*305  (^1-^)  +  (pi-p9) ^-r-,  (Ex.  1). 

(  —  J   is  the  limit  of  the  ratio  of  the  change  of  the  maximum  pressure  of  the 


Available  Energy.  507 

vapour  to  the  corresponding  change  of  temperature,  and  is  hence  easily  found 
from  a  table  of  the  temperatures  of  the  boiling-point  at  different  pressures, 
jffis  in  general  positive  ;  i.  e.  if  no  heat  be  imparted  to  the  expanding  steam, 
some  of  it  will  condense. 

7.  In  the  preceding  example,  if  the  vessel  from  which  the  steam  is  escaping 
contain  boiling  water,  determine  the  quantity  of  heat  which  must  be  imparted 
to  the  unit  mass  of  steam  in  order  that  it  should  remain  saturated. 

In  this  case,  as  in  Ex.  2,  Art.  338,  the  external  work  TTdone  by  the  unit 
mass  of  the  steam  is 

P2<r-2  -jtncri,  or povz  -piv\+p?,S2—piSi. 

Hence,  as  TJo  —  TJ\  is  the  same  as  in  the  last  example,  if  we  neglect  the  compa- 
ratively small  terms  involving  s\  and  so,  we  have  approximately 


-£(-*») 


and  therefore  E  =  -  0  •  30 5  ( Tx  -  To) . 

In  this  case,  if  no  heat  be  abstracted,  or  imparted,  the  steam  after  it  escapes 
is  super-heated.  If  T\  —  To  be  large,  or  the  steam  originally  at  high  pressure, 
the  super-heating  is  considerable  and  more  than  sufficient  to  vaporize  any  par- 
ticles of  water  which  the  steam  carries  with  it  mechanically.  Hence  we  can 
explain  the  known  phenomenon  that  high  pressure  steam  after  escaping  into  the 
air  is  dry  and  does  not  scald,  whereas,  by  low  pressure  steam,  severe  scalds  may 
be  inflicted. 

340.  Available  Energy. — The  work  which  can  be 
accomplished  by  a  quantity  of  heat  Qi  depends  on  the  tem- 
perature of  the  source  from  which  it  is  derived.  If  this 
temperature  be  TXi  and  the  lowest  temperature  which  can  be 
obtained  T0,  the  work  which  can  be  accomplished  by  means 

of  Qi  cannot  exceed  (Tx  -  T0)  -^  (Art.  338),  where  Qx  is  ex- 

J- 1 
pressed  in  work  units. 

If  Qv  pass  from  a  source  at  Tx  to  a  source  at  T2,  the  avail- 

(  T      T  \ 
able  energy  is  diminished  by  the  quantity  («?  -  t^)^1* 

If  Qx  leave  a  source  at  Tlt  and  Q>  in  consequence  enter  a 
source  at  To,  the  loss  of  available  energy  is 

(T,  -  To)  |  -  (21  -  Jl)  |2,  or  Qt-  ft-  T0 (|  -  -|). 


508  Thermodynamics. 

341.  Dissipation  of  Energy. — If  the  transference  of 
heat  from  a  source  at  Ty  to  a  source  at  T2  take  place  through 
the  medium  of  a  reversible  engine  undergoing  a  cyclical 

process,  ~ ■  -  -£  is  zero,  and  the  loss  of  available  energy  is 

Q\  -  Q2,  which  is  the  same  as  the  work  done.     Thus  the  un- 
compensated loss  of  available  energy  is  zero. 

In  the  case  of  an  engine  undergoing  a  non-reversible 
cyclical  process,  Ql  -  Qz  cannot  be  greater,  and  is  usually  less, 

than  (2\  -  Tt)  %  (Art.  337),  or  ~  -  %  has  a  negative  value 

1  \  J- 1      Jj 

which  may  be  denoted  by  -  N.  In  this  case  the  uncompen- 
sated loss  of  available  energy  is  T0  N. 

By  a  method  similar  to  that  employed  in  Art.  331  this 
result  can  be  extended  to  every  non-reversible  cyclical  pro- 
cess.     In  this   case,   if  Q  be   the  heat  which    enters   the 

Q  . 
engine  at  the  temperature  T,  the  quantity  S  -~  is  negative, 

and  the  uncompensated  loss  of  available  energy  is  -  T0E  -=. 

To  prove  this,  we  have  only  to  substitute  for  the  actual 
process  A  a  process  B  in  which  the  cycles  corresponding  to 
each  pair  of  temperatures  are  completed  by  reversible  trans- 
formations, each  of  which  is  accomplished  first  in  one  direc- 
tion, then  in  the  opposite.   As  these  transformations  are  passed 

through  in  both  directions,  the  value  of  2  —  and  of  the  un- 
compensated loss  of  available  energy  is  the  same  for  A  as 
for  B ;  but  2  -=  f or  B  is  the  sum  of  the  values  of  S  -j,  corre- 
sponding to  the  small  cycles,  since  the  remaining  part  of  B 
forms  one  reversible  cycle.  Hence  we  obtain  the  required 
results. 

The  uncompensated  loss  of  available  energy  is  called  the 
Dissipation  of  Energy. 

From  the  present  and  preceding  Articles  it  appears  that 
this  dissipation  takes  place  whenever  heat  passes  without  the 
performance  of  work  from  a  body  at  a  higher  to  a  body  at  a 
lower  temperature,  and  also,  in  general,  in  non-reversible 


Increase  of  Entropy.  509 

cyclical  processes.  A  strictly  reversible  process  cannot  be 
realized  in  nature,  since  the  absence  of  friction  and  the  perfect 
equality  of  internal  and  external  pressures  and  temperatures 
cannot  be  attained.  Hence  we  may  conclude,  that  in  natural 
processes  there  is,  in  general,  an  incessant  dissipation  of 
energy. 

There  is  one  class  of  irreversible  transformations  in 
which,  according  to  Mr.  Parker  (Philosophical  Magazine, 
June,  1888),  there  is  no  dissipation  of  energy.  Mr. 
Parker  in  the  Article  referred  to  defines  an  equilibrium 
path  to  be  one  at  every  point  of  which  the  system  is  in 
equilibrium.  The  path  corresponding  to  a  reversible  trans- 
formation is  always  an  equilibrium  path,  but  an  equilibrium 
path  is  not  necessarily  reversible.  As  a  result  of  experiments 
on  the  solubility  of  various  substances,  Mr.  Parker  has  been 
led  to  adopt  the  conclusion  that  in  an  irreversible  equilibrium 
cycle  there  is  no  dissipation  of  energy. 

It  is  to  be  observed  that  the  theory  of  dissipation  depends 
on  the  assumption  of  a  certain  temperature  as  the  lowest 
which  is  available.  If  the  lowest  available  temperature 
were  absolute  zero  there  would  be  no  dissipation  of  energy. 

342.  Increase  of  Entropy. — If  an  element  of  heat  rlQ 
pass  from  a  body  A,  whose  temperature  is  Ti9  to  another 
body  B  at  a  lower  temperature  T2,  and  if  we  suppose  the 
volumes  of  A  and  B  to  remain  constant,  the  entropy  of  A  is 

diminished  by  7=-,  and  that  of  B  increased  by  — ,  and  as 
1\  1% 

Tx  >  To,  the  whole  entropy  of  A  and  B  is  increased. 

Again,  in  a  cyclical  process,  if  we  suppose  the  source  A 

and  the  condenser  B  to  remain  at  constant  volume,  in  which 

case  their  temperatures  will  of  course  vary,  2  ~  is  the  loss 

of  entropy  by  A,  and  2  -^  the  gain  of  entropy  by  B. 
Hence  the  entropy  of  the  whole  system  is  increased  by 
the  quantity  2  (  ^  -  -^  ).  In  a  reversible  process  this  quan- 
tity is  zero,  but  in  a  non-reversible  process  it  has  in  general 
a  positive  value  N. 


510  Thermodynamics. 

"We  have  supposed  A  and  B  to  remain  at  constant  volume ; 
but  if  this  be  not  the  case,  the  results  obtained  still  hold  good, 
provided  the  transformation  applied  to  each  of  these  bodies  is 
reversible  when  each  body  is  considered  alone.  Under  these 
circumstances  the  uncompensated  loss  of  available  energy  in 
a  non -reversible  cyclical  process  is  equal  to  the  product  of 
the  limiting  temperature  and  the  increase  of  the  entropy  of 
the  system. 

Since,  according  to  Mr.  Parker,  there  is  no  dissipation  of 
energy  in  an  equilibrium  cycle  even  though  it  be  irreversible, 
in  such  a  cycle  the  entropy  of  the  whole  system  is  constant. 
Again,  it  would  appear  that  the  definition  of  entropy  in 
Art.  332  is  unnecessarily  restricted,  and  that  entropy  may 

—  along  any  equilibrium  path. 

It  would  seem  that  the  result  of  Mr.  Parker's  experi- 
ments might  have  been  anticipated.  For,  when  a  system  under- 
goes a  transformation  corresponding  to  an  equilibrium  path, 
the  irreversibility  of  the  transformation  for  the  whole  system 
can  result  only  from  the  way  in  which  heat  is  communicated 
to  or  leaves  the  system,  or  on  the  mode  in  which  it  passes 
from  one  part  of  the  system  to  another  part.  We  may 
therefore  suppose  the  system  divided  into  portions  for  each 
of  which  taken  separately  a  reversible  path  may  be  assigned 
coinciding  with  the  actual  equilibrium  path.  If  Qif  Q2,  &c. 
be  the  quantities  of  heat  which  at  any  stage  of  the  transfor- 
mation have  passed  into  these  portions,  Ul9  U2,  &c  their 
energies,  vl9  i\,  &c.  their  volumes,  pl9  p2,  &c.  their  pressures, 
Tl9  1\,  &c.  their  temperatures,  and  01?  <£2,  &c.  their  entropies, 
we  have  dQx  =  dUi  +  pidi\  =  Tx  dfr,  since  the  path  coincides 
with  a  reversible  path.     In  like  manner 

dQ2  =  dU2  +  p2dv2  =  T2  d<j>2,  dQ3  =  dU3  +p3  dv?J  =  T3dfa,  &c. 
Now,  since  the  whole  system  is  in  equilibrium, 

rl=  T2  =  T3  =  &c.  =  T,  pi=p2  =p3  =  &c.  =p. 


Path  of  Lead  Heat.  511 

Hence,  if  0  be  the  entropy  of  the  entire  system,  and  Q  the 
quantity  of  heat  imparted  to  it, 

p         dQx  +  dQv+ &g.     dQ 
d$  =  dfa  +  dfa  +  &c.  =  -= =  -jr, 

and  therefore  so  far  as  the  relation  between  heat  imparted 
and  entropy  is  concerned,  the  whole  transformation  may  be 
treated  as  if  it  were  reversible. 

We  may  conclude  from  what  has  been  said,  that  natural 
processes  have  a  tendency  to  increase  entropy,  or,  as  stated 
by  Clausius,  the  entropy  of  the  universe  tends  to  become  a 
maximum. 

343.  Path  of  Least  Bleat. — Let  us  suppose  that  a 
body,  whose  entropy  is  fa,  passes  from  the  state  A  to  the 
state  B  in  which  its  entropy  is  fa,  less  than  fa.  If  Q  be 
the  heat  given  out  by  the  body  when  at  the  temperature 

Q 

T,  and  if  S  denote  the  value  of  S  —  for  the  whole  process, 

S  cannot  be  less  than  fa  -  fa.  To  prove  this,  first  suppose 
the  transformation  reversible,  then  8=  fa-  fa.  Next  suppose 
the  transformation  non-reversible,  and  let  the  cycle  be  com- 
pleted by  a  reversible  process  which  brings  the  body  from 

B  to  A.     The  value  of  2  -p=,  for  the  cycle  is  then  S-  (fa-  fa), 

and  this  must  be  positive  (Art.  340)  ;  hence  8  >  fa  -  fa. 

Let  us  now  consider  by  what  path  a  body,  whose  tempe- 
rature can  never  be  less  than  T0f  should  pass  from  the  state 
A  to  the  state  B  at  T0,  fa,  so  that  the  heat  given  out  in  the 
passage  should  be  a  minimum,  no  heat  being  supplied  to  the 
body  from  any  external  source. 

Let  H  be  the  heat  given  out ;  then  for  a  non-reversible 
transformation,  since  T  >  T0,  and  since  any  element  of  heat 
which  enters  the  body  at  T  must  have  previously  passed 
out  of  it  at  a  temperature  higher  than  T,  we  must  have 
H  >  TV)S  >  T0 ((pi  -  fa).      For  a  reversible  transformation 

Tdfa  which  is  least  when  T  =  T0.     The  least  value  of 

tf>0 

H  is  therefore  T0(fa  -  fa).  Hence  the  path  consists  of  an 
adiabatic  at  the  entropy  fa  from  Tx  to  T0,  and  an  isothermal 


512  Thermodynamics. 

at  To  from  fa  to  fa.  Since  JTi  -  Z7"o  =  TF+  #,  where  TFis  the 
work  done  by  the  body  during  the  transformation,  when  H 
is  least  W  is  greatest,  and  the  maximum  work  which  a  body 
can  perform  under  the  circumstances  supposed  is 

Ui-UQ-TAfa-fa). 


Examples. 
1.  Prove  that  the  available  energy  of  any  system  of  bodies  is 


J  T0 


where  T\  is  the  initial  temperature  of  mi,  and  c\  its  specific  heat  at  constant 
volume. 

2.  If  the  system  in  Ex.  1  be  enclosed  in  an  envelope  impermeable  by  heat, 
show  that  To  is  determined  by  the  equation 

>ri      dT 


I,.*-?-0- 


The  actual  work  performed  by  the  system  during  the  transformation  in 
which  all  its  parts  are  brought  to  the  temperature  To  is 


]T0 


P  CidT; 

J  T0 


but,  if  the  transformation  be  that  in  which  the  greatest  possible  work  is  done, 
this  work  must  be  equal  to  the  available  energy,  and  therefore 


2mi 


When  the  limiting  temperature  To  is  determined  from  within,  as  in  this  ex- 
ample, or,  in  other  words,  when  one  part  of  the  system  acts  as  condenser  to 
another  part,  the  available  energy  is  called  by  Thomson  the  Internal  Thermo- 
dynamic Motivity.  When  To  is  independent  of  the  system,  i.e.  when  heat  can 
pass  out  of  the  system  to  an  external  condenser,  the  available  energy  may  be 
termed  the  External  Thermodynamic  Motivity.     In  this  case  To  must  be  as- 


3.  If  a  system  consist  of  two  equal  masses  of  the  same  substance  whose 
specific  heat  is  constant,  show  that  the  limiting  temperature  of  the  internal 
thermodynamic  motivity  is  \ '  T1T2,  where  Ti  and  T2  are  the  initial  temperatures 
of  the  two  masses. 

4.  In  the  preceding  example  prove  that  the  thermodynamic  motivity  of  the 
system  is  me  (V ~Ti  -  Vl^)2. 


Examples.  513 

5.  If  the  entropy  of  a  substance  be  increased,  its  energy  remaining  constant, 
prove  that  the  work  which  can  be  obtained  by  a  transformation  to  a  given  state 
is  diminished. 

6.  A  unit  mass  of  gas,  whose  volume  is  v\,  is  allowed  to  expand  into  a  per- 
fectly empty  vessel,  whereby  its  volume  becomes  v%  •  show  that  its  capability 

of  doing  work  is  diminished  by  the  quantity  TaR  log—. 

V\ 

7.  Determine  a  transformation  by  which,  without  the  transference  of  any 
heat,  gas  at  p\V\  may  be  brought  by  the  application  of  the  smallest  possible 
amount  of  external  work  to  p>2V% ;  where  pi  >  p\,  vi  >  V\. 

Since  »2  >  »i  the  gas  must  expand,  and  since  no  heat  is  given  it  must  ex- 
pand by  its  own  energy.  It  will  do  this  with  the  smallest  possible  expenditure 
of  energy  by  expanding  into  a  vacuum.  If  TT%  be  the  energy  corresponding  to 
P2V2,  the  smallest  amount  of  external  work  capable  of  changing  the  energy  from 
U\  to  Ui  is  Z7o  —  U\,  and  in  order  that  no  more  than  this  should  be  required  the 
compression  must,  by  (25),  be  adiabatic.  Hence  let  the  gas  expand  into  a 
vacuum  till  its  volume  become  v,  and  then  let  it  be  compressed  adiabatically 
till  its  volume  become  v%.  In  order  to  determine  v,  let  T\  and  To  be  the  tem- 
peratures belonging  to  the  initial  and  final  state ;  then,  by  (16), 


T\vk~x  =  T2V2*"1,     whence 


■0-- 


2  L 


514  Miscellaneous  Examples. 


Miscellaneous  Examples. 

1.  If  two  points  fixed  in  a  lamina  slide  upon  two  intersecting  straight  lines, 
and  if  one  point  be  made  to  oscillate  backwards  and  forwards  so  as  to  have  always 
the  same  velocity,  the  ellipse  described  by  any  fixed  point  of  the  lamina  will  be 
described  under  acceleration  which  is  fixed  in  direction. 

2.  A  material  point  of  given  mass  moves  freely  under  the  action  of  a  central 
force  of  given  absolute  intensity,  varying  inversely  as  the  square  of  the  distance ; 
given  the  initial  circumstances  of  projection,  determine  the  major  axis,  eccen- 
tricity, and  line  of  apsides  of  the  orbit  it  describes. 

3.  The  extremities  of  a  uniform  rectilinear  bar  move  on  the  circumference 
of  a  smooth  vertical  circle ;  find  its  period  of  oscillation  under  the  action  of 
gravity  consequent  on  a  small  displacement  from  its  position  of  stable  equi- 
librium. 

4.  A  circular  plate,  revolving  round  its  centre  in  a  vertical  plane,  becomes 
suddenly  attached  at  its  lowest  point  to  a  heavy  particle  previously  at  rest ;  re- 
quired the  mass  of  the  particle  in  order  that,  at  the  end  of  a  semi-revolution,  the 
system  may  be  brought  to  rest  under  the  action  of  gravity. 

5.  A  uniform  beam  is  supported  symmetrically  on  two  props  ;  find  where 
they  should  be  placed  in  order  that  if  one  of  them  be  removed  the  instantaneous 
pressure  on  the  other  may  be  the  same  as  the  statical  pressure. 

6.  A  circular  board  lies  upon  a  smooth  table  ;  in  the  board  is  cut  a  circular 
groove  along  which  a  molecule  is  projected  with  a  given  velocity ;  determine 
the  pressure  against  the  side  of  the  groove. 

7.  A  straight  rod  which  passes  through  a  small  fixed  ring  is  in  motion  in  a 
horizontal  plane  ;  determine  the  motion  of  its  centre  of  gravity. 

8.  A  lamina  unacted  on  by  any  force  is  projected  in  its  own  plane ;  prove 
that  its  space  centrode  is  a  straight  line,  and  its  body  centrode  a  circle. 

9.  A  sphere,  rotating  about  a  horizontal  axis  through  its  centre  of  gravity, 
falls  vertically ;  prove  that  its  space  centrode  is  a  parabola,  and  its  body  cen- 
trode a  spiral  of  Archimedes. 

10.  Given  the  motion  of  one  point  in  a  body  and  also  its  space  centrode,  find 
its  body  centrode. 

11.  A  small  ring  slides  down  a  rough  rod  from  a  given  point  to  a  given 
right  line ;  find  the  direction  of  the  rod  so  that  the  time  of  descent  may  be  a 
minimum. 

(a)  Find  the  limits  of  the  coefficient  of  friction  for  which  the  required  posi- 
tion is  vertical. 


Miscellaneous  Examples.  515 

12.  A  material  particle,  attached  to  a  fixed  point  by  an  inelastic  string,  is 
allowed  to  descend  a  smooth  inclined  inelastic  plane,  starting  without  initial 
velocity  from  the  foot  of  the  perpendicular  from  the  fixed  point  on  the  plane. 
Describe  the  subsequent  motion,  and  show  that  the  total  length  of  the  path 
described  by  the  particle  on  the  plane  before  it  comes  to  rest  is 


\sm 


sin/3  cos2/3 
l  +  cos'-jS 


where  I  is  the  length  of  the  string,  and  j3  is  the  angle  which,  when  stretched, 
it  makes  with  the  perpendicular. 

13.  A  homogeneous  sphere  rolls  down  the  concave  surface  of  a  rough  semi- 
circle, the  axis  of  which  is  vertical ;  find  its  velocity  and  entire  pressure  against 
the  semicircle  in  any  position. 

14.  Two  balls  of  different  masses,  moving  in  the  same  right  line  with  diffe- 
rent velocities,  become  suddenly  connected  by  a  weightless  inextensible  rod : 
given  all  particulars,  required,  in  magnitude  and  direction,  the  initial  strain  on 
the  rod. 

15.  A  material  particle,  constrained  to  oscillate  without  friction  in  a  curve 
tautochronous  with  respect  to  any  point  under  the  action  of  any  force,  being 
supposed  retarded  throughout  its  motion  by  a  resistance  to  its  velocity  of  con- 
stant intensity  ;  determine  the  law  of  diminution  of  its  several  successive  arcs 
of  vibration. 

16.  The  resisting,  in  the  preceding,  being  supposed  small  compared  with  the 
moving  force ;  show  that,  if  the  friction  vary  as  any  function  of  the  velocity, 
its  effect  will  be  ultimately  inappreciable  on  the  time  of  description  of  any  com- 
plete arc  of  vibration  of  the  particle. 

17.  A  rigid  body,  revolving  round  a  fixed  axis,  strikes  perpendicularly 
against  a  fixed  obstacle;  required  the  height  through  which  the  same  body 
should  fall  vertically,  without  rotation,  so  as  to  strike  against  the  obstacle 
with  the  same  force  of  percussion. 

18.  A  rigid  body  connected  with  a  fixed  point  by  an  inextensible  cord,  is  in 
constrained  equilibrium  under  the  action  of  a  force  passing  through  its  centre 
of  inertia ;  all  the  other  restraints  being  supposed  suddenly  removed,  required 
the  initial  stress  on  the  cord. 

19.  A  sphere,  rolling  without  sliding  on  a  rough  horizontal  plane,  is  acted 
on  by  a  central  force,  varying  inversely  as  the  square  of  the  distance,  emanat- 
ing from  a  fixed  point  in  the  parallel  plane  passing  through  its  centre.  Show 
that  it  describes  a  focal  conic  round  the  centre  of  force ;  and  determine  the 
initial  velocity  for  which  the  motion  is  parabolic. 

20.  A  rigid  body,  being  set  in  motion  by  a  single  impulsive  force,  show  that 
all  axes  of  initial  pure  rotation,  corresponding  to  different  directions  of  the  per- 
cussion, envelope  a  quadric  cone,  diverging  from  the  centre  of  inertia,  and 
touching  the  three  central  principal  planes  of  the  body. 

2L  2 


516  Miscellaneous  Examples. 

21.  A  rigid  body,  having  two  fixed  points,  is  set  m  motion  by  an  impulsive 
force ;  determine  in  magnitude  and  direction  the  initial  percussions  at  the  points 
perpendicular  to  their  line  of  connexion. 

22.  Two  material  particles,  resting  on  a  rough  inclined  plane,  and  connected 
by  a  slight  flexible  cord,  passing  without  friction  through  a  small  ring  attached 
to  a  fixed  point  on  the  plane,  are  in  equilibrium  under  the  action  of  gravity  ; 
the  inclination  of  the  plane  being  supposed  gradually  increased,  or  its  roughness 
ness  gradually  diminished,  determine  the  nature  of  the  initial  motion  of  the 
particles. 

23.  Two  material  particles,  moving  without  friction  in  two  non-intersecting 
rectilinear  tubes  of  indefinite  length,  attract  each  other  with  a  force  varying 
directly  as  their  distance  asunder ;  determine  completely  their  motion. 

24.  In  the  general  displacement  of  a  solid  from  one  given  position  to  another, 
find,  by  geometrical  construction,  the  twist  by  which  the  body  can  be  brought 
from  the  former  to  the  latter  position. — (Prof.  Crofton,  London  Mathematical 
Society,  1874.) 

Let  A  be  any  point  of  the  solid  in  its  first  position,  B  the  new  position  of 
the  same  point ;  again,  let  C  be  the  new  position  of  the  point  which  was  origi- 
nally at  B,  and  D  the  new  position  of  that  point  originally  at  C;  then,  to  find 
the  required  twist,  bisect  the  angles  ABC  and  BCD  by  the  lines  BBTand  CK; 
find  EKthe  shortest  distance  between  these  bisectors.  The  body  can  be  brought 
from  the  first  to  the  second  position  by  a  translation  HK,  and  a  rotation  round 
HK  through  an  angle  which  is  equal  to  that  between  BIT  and  CK. 

25.  Calculate,  in  C.  G.  S.  units,  the  mutual  attraction  of  two  units  of  mass 
at  the  unit  distance  apart,  according  to  the  law  of  gravitation. 

Let  7  denote  the  quantity  in  question ;  then  the  attraction  of  the  earth  on  a 
unit  of  mass  at  its  surface  is  %irypB,  where  p  is  earth's  mean  density,  and  It  is 
its  radius. 

Hence  we  have  g  =  iirpylt. 

Now,  in  the  system  of  units  adopted,  we  have  g  =  981,  and  irlt  =  2  x  109. 
Thence,  assuming  p  =  5-67,  we  get 

I  =  ?  x  551  x  107  =  —  x  107  =  15,410,000,  approximately ; 
7      3      981  109 

•••     ?  =  15,410,000  drneS' 

26.  A  body  is  rotating  about  a  fixed  point.  Express  the  element  of  the  curve 
described  by  the  instantaneous  axis  on  a  sphere  fixed  in  the  body  in  terms  of  the 
angular  velocities  round  the  body-axes. 

Let  the  instantaneous  axis  at  any  time  make  angles  A,  /j.,  v  with  the  body- 
axes  ;  let  the  spherical  surface  be  intersected  by  the  two  consecutive  positions  of 
the  instantaneous  axis  in  J  and  /' ;  let  OJand  OI"  represent  the  correspond- 
ing magnitudes  «  and  w  +  rfwof  the  angular  velocity.  Then  the  projections  of 
II"  on  the  body-axes  are  proportional  to  du\,  da2,  decs,  and  II"  is  propor- 
tional to  du. 


Miscellaneous  Example 


517 


Now  II"2  =  IT2  +  OFdy* ; 

hence  dwi2  +  do>22  +  door  =  dot2  +  ordty2, 

and  dip2  =  —  (^a>i2  +  dcez2  +  ^a>32  -  d<a2). 


If  «  be  the  radius  of  the  sphere,  and  <r  the  arc  of  the  curve, 
we  have,  therefore, 


(SM8 


1(5 


\2       /^o\2      /^3\2      /<?»\2 

)+(*H*H*) 


27.  A  body  is  moving  round  a  fixed  point.  Being  given  the  axis,  a  rotation 
round  which  brings  the  body  from  one  position  to  another,  and  the  magnitude 
of  the  rotation,  determine  the  angles  which  body-axes  make  in  the  second  posi- 
tion with  the  space-axes  which  in  the  first  position  coincide  with  them. 

Describe  a  sphere  round  the  fixed  point  0.     Let  two  of  the  space-axes  meet 


this  sphere  in  the  points  X,  T;  and  the  corresponding  body-axes  in  the  points 
A,  B,  when  the  body  is  in  its  second  position  ;  let  P  be  the  pole  of  rotation ; 
then  XPA  =  YPB  =  <p,  where  <p  is  the  given  rotation.  Let  I,  m,  n  be  the  direc- 
tion cosines  of  the  angles  that  OP  makes  with  OX,  OY,  OZ. 

Then  cos  XA  —  P  +  (1  -  I2)  cos  <p,  and  it  can  be  readily  shown  that 
cos  YA  =  Im  (1  -  cos  cp)  +  n  sin  <p, 
and  cos  XB  =  Im  (1  -  cos  <J>)  —  n  sin  </>, 

The  values  of  the  cosines  of  the  remaining  angles  can  now  be  written  down 
from  symmetry. 
If  we  put 

v  =  cos  7f  <p,     A  =  I  sin  h  <j>,     fj.  =  m  sin  §  <p,     v  =  n  sin  \  <p, 

we  have  the  following  table  of  the  values  of  cos  XA,  &c. : — 


X 

Y 

z 

A 

vl  +  A2  —  /jl2  —  v2 

2(/uA  +  vv) 

2  (v\  -  vfi) 

B 

2  (\fi  -  w) 

V2  +  fM2  —  V2  —  A2 

2{vfi+  v\) 

C 

2  (\v  +  v,u) 

2  (fjLV  -  vA) 

v2  +  v2  -  A2  - 

The  quantities  I,  m,  n,  <p  are  called  Rodrigues'  coordinates.     (Thomson  and 
Tait,  Natural  Philosophy,  §95.) 


518 


Miscellaneous  Examples. 


28.  The  body  is  rotating  round  OA  with  an  angular  velocity  w\ ;  determine 
the  differential  coefficients  of  <p,  I,  m,  n  with  respect  to  the  time. 


"We  have  to  find  the  magnitude  and  axis  of  the  rotation  which  is  the  resultant 
of  the  rotation  $  round  the  axis  (I,  m,  n),  and  of  on  dt  round  OA.     If  this  rotation 

be  <$>',  and  its  direction  cosines  I',  m',  ri  ;  <£>'  -  <p  =  —  dt,     V  -  I  =  —  dt,  &c. 

Let  A,  B,  Cbe  the  points  in  which  OA,  OB,  OCmeet  the  sphere  described 
round  0,  and  P  the  point  in  which  the  sphere  is  met  by  the  line  (I,  in,  n). 
As  in  Ex.  8,  Art.  260,  make  APE  =  if,  and  PAP  =  -  |  on  dt,  then  P  is  the 
pole  of  the  resultant  rotation  ;  the  positive  direction  of  rotation  being  supposed 
to  be  counter-clockwise. 

Draw  R  V  at  right  angles  to  AP.     Then  it  is  easily  seen  that 


cos  B PA  = 


fan 


V{(1- ?)(!-»**)}' 


sin  BPA  = 


V{(l-J2)(l-™2)}' 


also 


2  —  =  a>i  sm2^P  — ^— , — g=  m  1  -  J2)  cot  J  tf>, 
eft  smid>  * 


Again 


2  —  =  wi  sin  ^P sm  PP— r— r —  =  «i  (w  -  fo»  cot  f  0), 


o)i  sin  AP  sin  CP 


sini(/> 
cos  CPP 


BOO.  ftp 

\<p'=Tt-APP, 


=  -  a»i  (w  +  nl  cot  J 0) . 


and 


o)\dt  (p  0 

cos  ARP  =  — --  cos  AP  sm  -  -  cos  - 


hence 


cos  \  <p'  =  cos  J  #  -  f  foi  sin  J  ^^  ; 


dtp 

dt 


=  foi 


Miscellaneous  Examples.  519 

29.  A  body  is  moving  round  a  fixed  point  0,  with  angular  velocities  a>i,  «2,  a>3, 
round  three  rectangular  axes  OA,  OB,  00  fixed  in  the  body.  Determine  the 
differential  coefficients  of  Rodrigues'  coordinates  with  respect  to  the  time. 

By  means  of  the  last  example  we  can  write  down  the  changes  produced  on 
</>,  I,  m,  n  by  each  of  the  rotations  co\dt,  w^dt,  oizdt. 
Adding,  and  dividing  by  dt,  we  get 

2  —  =  —  (aon  +  W3«J  4-  COt|(J>{a>i  -  l(lw\  +  w«2  +  nooz)}, 
dt 

2  —  =  -  wil  +  w\n  +  eoth<p{u2  —  m{lm  +  muz  +  nm)}, 
dt 

2  —  =  -  wi»  4-  a>2 1  +  cot |0{«3  -  n(lu)\  +  ma>2  +  na>z)}, 
dt 

d<p      , 

■—  =  l(i}\  +  mw%  +  wa>3 ; 

dt 

whence,  also,  we  obtain 

ndv  nd\ 

2  —  =  -  <)>l\  -  W2H  -   WiV,        Z—  =  CCIV  —  C02V  +  003 fJ., 

dt  dt 

du  .  dv 

2  =  OJOV  —  CCZ\  +  OJ\V,  2  —  =  0>3U  -  Wljll  +  o>oA, 

«£  at 

where  v,  A,  /*,  v  have  the  same  meaning  as  before. 

30.  A  rigid  body  is  moving  in  any  manner ;  one  point  is  suddenly  arrested ; 
determine  the  impulse  exerted  on  the  body. 

Let  u,  v,  w  be  the  components  of  the  velocity  of  the  point  immediately  before 
it  is  arrested,  x,  y,  z  its  coordinates,  and  X,  Y,  Zthe  components  of  the  impulse, 
the  axes  being  the  principal  axes  of  the  body  at  the  centre  of  inertia,  then  X  is 
given  by  the  equation 

-  [^-  +  A(B  +  C)x>  +  B(C+A)y°-  +  C(A  +  J5)z2  +  mir^X 

=  {ABC  +  ®l[A{B  +  C)xi  +  BC(y2  +  z2)  +  BtIt*sP\}« 

+  Tl{AB  +  Wilr°~)xyv  +  Wl(AC  +  2)llr2)xzw, 

where  i"  is  the  moment  of  inertia  of  the  body  round  the  line  joining  the  arrested 
point  to  the  centre  of  inertia,  r  the  distance  between  these  points,  and  A,  B,  C, 
the  principal  moments  of  inertia  of  the  body. 

31.  A  sphere  is  projected  in  any  way  along  an  imperfectly  rough  inclined 
plane.     Investigate  the  motion. 

(This  investigation,  with  some  slight  modifications,  is  taken  from  Routh, 
Rigid  Dynamics.) 


520  Miscellaneous  Examples, 

Here  the  equations  of  motion  are 

Mx  =  X  4-  Mg  sin  i,     My  =  Y, 

§  Mr-wi  =  rY,     %Mr2d>o  =  -  rX, 

whence,  eliminating  X  and  Y,  we  obtain,  on  integrating, 

z  +  %r<a2  =  fft  sini  +  a  +  f rfl2, 

//  -  %ra>i  =  &  -frHi, 

where  a,  )8,  Hi,  and  Ho  are  the  initial  values  of  x,  y,  «i,  and  «2. 

Again,  if  u  be  the  velocity  at  any  instant  of  that  point  of  the  sphere  which 
is  in  contact  with  the  plane,  and  9  the  angle  which  its  direction  makes  with  the 
axis  of  x, 

u  cos  9  =  x  -  r«2,     u  sin  9  =  y  +  ru\. 

Differentiating,  substituting  for  x,  &c,  from  the  equations  of  motion,  put- 
ting for  X  and  Y  the  values  which  they  take  as  long  as  there  is  slipping,  viz., 
-  [xMg  cos  i  cos  9  and  —  fiMg  cos  i  sin  9,  and  solving  the  resulting  equations  for 
u  and  u9,  we  have 

u  =  g  sin  %  cos  9  —  ^fxg  cos  i,     u9  =  —  g  sin  i  sin  9. 

Hence,  if  |/a  cot  i  =  n,  we  obtain,  by  integration,  u  sin  9  =  K\  (tan  J0)". 
Substituting  the  value  given  by  this  equation  for  u  in  the  equation  for  u9,  and 
integrating,  we  have 

(tan|fl)»+1      (tanp)"'1  =  ^  _  tysini  ^ 
w  +  1  n  -  1  2  -fiT2       ' 

A^  is  determined  from  the  initial  value  of  9,  and  K\  from  the  initial  values 
of  9  and  w.     These  latter  are  given  by  the  equations 

uQ  cos  0o  =  a  —  rfc,      u0  sin  9q  =  £  +  rfli  ; 

then  «*  and  9  being  known,  #,  y,  o>i,  and  &>2  can  be  determined. 

If  n  or  f  fx  cot  i  >  1,  u  and  0  become  continually  less  until  they  vanish 
together.     Pure  rolling   then   begins   at  a   time    t0,  which  is   given  by  the 

equation  to=  - — r— :•     After  pure  rolling  begins  the  values  of  x,  y,  u\,  and  wz, 

at  any  time,  can  be  obtained  from  the  combination  of  the  equations  of  motion 
with  the  equations 

x  —  r«2  =  0,     y  +  rw\  =  0. 

If  n  <  1,  9,  though  constantly  approaching  zero,  as  appears  from  the 
expression  for  u9,  will  not  vanish  in  any  finite  time,  and  u  tends  to  increase 
without  limit. 


Miscellaneous  Exercises.  521 

If  u0  =  0,  the  problem  is  at  starting  reduced  to  that  of  Ex.  3,  Art.  278. 
The  force  of  friction  requisite  for  pure  rolling  is  then  §■  Mg  sin  i.     Hence,  if 

f  Mg  sin  i  <  n'Mg  cos  i,     or     J  y!  cot  i  >  1 , 

where  yJ  is  the  coefficient  of  statical  friction,  pure  rolling  will  commence  and 
continue.     If  %/a  cot  i  <  1,  slipping  will  begin  at  once  and  never  cease. 

32.  A  body  rests  with  a  plane  face  on  an  imperfectly  rough  horizontal  plane. 
The  centre  of  inertia  of  the  body  is  vertically  over  the  centre  of  inertia  of  the 
face  and  very  near  it,  the  connecting  line  being  a  principal  axis  at  the  former 
point.  The  form  of  the  face  is  such,  that  its  radii  of  gyration  about  all  lines 
in  it  passing  through  its  centre  of  inertia  are  equal.  The  body  is  projected 
with  an  initial  velocity  of  translation  U,  and  an  initial  very  small  angular 
velocity  fl  round  a  vertical  axis  through  its  centre  of  inertia  :  determine  the 
motion. 

Take  the  initial  direction  of  translation,  and  a  horizontal  line  at  right  angles 
thereto  for  axes  of  x  and  y.  Let  u  and  v  be  the  components  of  the  velocity  of 
the  centre  of  inertia  of  the  body  at  any  time,  and  w  the  angular  velocity. 
Then,  x  and  y  being  the  coordinates  of  any  point  of  the  body,  and  £  and  77  its 
coordinates  referred  to  parallel  axes  through  the  centre  of  inertia, 

clx  dy 

-  =  u-v»,   #=?  +  *>. 

If  F  be  the  magnitude  of  the  whole  force  of  friction  at  any  point,  its  com- 
ponents X  and  Y  are  given  by  the  equations 

X  =  -  F H  ~  ^  —  =  -F  a  v 


V{^  +  77o>)2+(v  +  !a>)2} 


V  +  £co 

--■F—T-*  2-P- 


since  0,  £a>,  and  rjca  are  small  compared  with  u. 

Again,  if  S  be  the  area  of  the  plane  face,  the  magnitude  of  the  normal  re- 
action of  the  horizontal  plane  on  an  element  of  the  face  is  equal  to  <f>  (£,  ij)  dS, 
whence  F  =  n<p  (£,  77)  dS,  and,  since  5  =  constant,  f  0  (|,  77)  dS  =  my,  where  m  is 
the  mass  of  the  body.  Also  equations  (17),  of  Art.  267,  give  Gx  =  0,  Gy  =  0, 
since     az  =  0,     wy  =  0,     i  =0,    /  =  0. 

If  a  be  the  distance  of  the  centre  of  inertia  of  the  body  from  the  plane  face, 
and  <j>  (|,  r,)  =  R, 

Gy=tia$ItdS-fl%dS; 

therefore  J  B£dS  =  iimga. 

Assume  R  =  K+  eA,  where  iT  and  e  are  constants,  then 

fimya  =  K  j  £dS  +  e  f Af«fe,     but    f&S  =  0  ; 
therefore  6  must  be  small ;  also 

mg  =  KS  +  e  J  AdS. 


522  Miscellaneous  Exercises. 

Again,  &»  =  J"  Bi\dS  -  jua  f  V-^-  KdS  ; 

and,  since  the  second  member  of  Gx  is  zero,  q.p.,vre  have  J"  iSrj^/S  =  0.     Hence 
the  resultant  normal  reaction  passes  through  a,  point  on  the  axis  of  x. 
To  determine  the  motion  of  the  centre  of  inertia, 

m—  =  2X  =  -  n  J"  BdS  =  -  \tmg ; 

therefore  w=  JT-figt. 

dv  v  f  0  f  v 

Again  »»  —  =  2F  =  -  ^  -  \  RdS  -  n  -  \  R£dS  =  -  \nmg  -,  q.p. 

°  dt  u]  u  J  w 

hence  0  =  cu ;  and  since  »  =  0  when  u=  U,  c  =  0,  therefore  v  =  0. 
To  find  the  angular  velocity, 


mk2 


3F--^J«"--i<J«"* 


but  7  being  the  radius  of  gyration  of  the  plane  face,  J"  £2tf£  =  #y2,  and 

w&2  —  =  -  fxmgy2  -  ;  ?•  .P- 

therefore  a>  =  ft  (77)**  • 


INDEX. 


Absolute,  units,  54,  126. 

force  in  central  orbit,  175. 

force  suddenly  changed,  179. 

zero  of  temperature,  481,  499. 
Acceleration,  uniform,  12. 

variable,  15. 

total,  17. 

tangential  and  normal,  17. 

angular,  19. 

areal,  21. 
Acceleration-centre,  267,  340. 
Acceleration  of  rotation,  327. 
Action  and  reaction,  58. 
Adiabatic  curve,  483. 
Airy,  on  Earth's  density,  107. 
Ampere's  Cinematique,  5. 
Angular  velocity,  19,  95. 

of  a  body,  95. 
Apsides,  190. 
Apsidal  angle,  191,  215. 
Areas,  uniform  description  of,  164. 

accelerations  of,  244. 

for  principal  plane,  245. 
Attraction,  law  of,  92,  130,  147. 
Atwood's  machine,  60,  64,  138. 
Axes,  relation  between  rotations  round 
space  and  body,  327,  331. 

Ball.  Sir  R.  S.,  referred  to,  59,  334, 

338,  407,  474. 
Ballistic  pendulum,  271. 
Bertrand,  on  closed  orbits,  203. 

theorem  of,  231,  420. 
Billiards,  problem  in,  387. 
Body  axes,  330. 

motion  referred  to,  385. 
Breaking  weight  of  elastic  string,  158. 
Bresse,  on  acceleration,  268. 
Bonnet's  theorem,  208. 
Bordoni,  82. 
Brachystochrone,  435. 


Calculus  of  variations,  433. 
Canonical  form  of  equations  of  mo- 
tion, 431. 
Carnot,  S.,  cycle  of,  486. 

extended,  488. 

determination  of  function  of,  487, 
500. 
Central  forces,  90,  147,  164. 

potential  of,  129. 
Centres  of  oscillation  and  percussion, 

276,  277. 
Centre  of  inertia,  76. 

of  oscillation,  142. 

motion  of,  241. 

motion  relative  to,  242. 
Centrifugal  and  centripetal  force,  88. 

acceleration,  89. 

force  at  Earth's  equator,  91. 
Centrifugal  force,   resultant   for   ro- 
tating body,  96. 

in  pendulum,  118. 
Centrifugal  couple,  369. 

axis  of,  370,  374. 
Centrodes,  261. 

Change  of  state  of  a  body,  501. 
Circle  of  inflexions,  268. 
Circular,  motion,  84. 

orbits,  90. 

orbits  approximately,  194. 
Clausius,  on  energy  of  a  gas,  410. 

on  second  fundamental  principle 
in  thermodynamics,  485. 

on  entropy,  489. 

on  saturated  steam,  505. 
Coaxal  circles,  property  of,  120. 
Coefficient  of  restitution,  67. 
Collision,  of  spheres,  direct,  66. 

effect  on  energy,  235. 

oblique,  70. 

of  smooth  bodies,  379. 

of  rough  bodies,  3S0. 


524 


Index. 


Compound  pendulum,  141. 
Compression,  force  of,  67. 
Composition  of  velocities,  7,  257. 

of  rotations,  317,  321. 

of  twists,  334. 
Cone,  employed  graphically  in  rota- 
tion, 328. 
Conical  pendulum,  115,  224. 
Conservative  system  of  forces,    129, 

233,  397. 
Constrained  motion,  206,  241,  247. 
Coulomb,  on  dynamical  friction,  63. 
Couple,  of  rolling  friction,  311,  314. 

of  twisting  friction,  311. 

tending  to  break  moving  rod,  303. 
Curtis,  225. 
Cycle,  Carnot's,  486. 
Cycloid,  tautochronism  of,  115. 

is  curve  of  quickest  descent,  433. 
Cylindroid,  338,  339. 

D'Alembert's  principle,  59,  227,  228, 

417. 
applied  to  small  oscillations,  445. 
Darwin,  on  friction  of  tidal   action, 

408. 
Degrees  of  freedom,  254,  269. 
Disturbing  forces  in  focal  orbit,  188. 
Dyne,  54,  126. 

Earth,  atttaetion  of,  151. 

mean  density  of,  107. 
Efficiency  of  agents,  126. 

of  a  heat  engine,  499. 
Elasticity,  67,  302. 

in  collision,  334,  383. 
Elastic  strings,  155. 
Elasticity  and   expansion   of   a   sub- 
stance, 491. 
Ellipsoid  momental,  348. 

graphical  use  of,  348. 
Ellipsoid,  of  gyration,  349,  360,  371. 

of  equal  energy,  474. 

potential,  474. 

conjugate,  363. 
Energy,  59,  133,  396. 

.  potential  and  kinetic,  defined,  133. 

measure  of  kinetic,  133. 

equation  of,  136,  396,  402. 

in  thermodynamics,  478,  485. 

conservation  of,  397. 

of  initial  motion,  420. 

of  an  oscillating  system,  470. 


Entropy,  489. 
Erg,  126. 

Euler,    equations    of    rotation,    354, 
368,  425. 
for  impulses,  346. 

Focal  orbit,  173. 

velocity  in,  177. 

constructed,  178. 
Force,  function,  398. 

measure  of,  53. 

absolute  unit  of,  54. 

gravitation  unit  of,  54. 
Forces  of  inertia,  59,  227. 
Fly-wheel,  energy  of,  137. 
Free  motion  of  a  body,  320. 
Freedom,  degrees  of,  254,  269. 
Friction,  laws  of  dynamical,  50,  63, 
296. 

work  expended  on,  in  pivot,  132. 

rolling,  311. 

twisting,  311. 

impulsive,  308. 

Gauss,  absolute  unit  of  force,  54. 
Generalized  coordinates,  415,  452. 

equations  of  motion,  421. 

impulse  components,  418. 
Geometrical   representation    of   rota- 
tion, 321. 
Goodeve,  265. 
Gravitation,  units,  54,  126. 

law  of,  176. 

verified,  93. 
Gravity,  acceleration  due  to,  29. 

variation  of,  30. 

affected  by  Earth's  rotation,  92. 

determined   by   pendulum,     102, 
143. 
Greenhill,  83. 
Gyration,  radius  of,  141. 

ellipsoid  of,  349. 


Hamilton's  equation  of  motion,  431. 

characteristic  function,  439. 
;   Harmonic  motion,  simple,  85. 

elliptic,  86. 
Harmonic  determinant,  454. 

real  roots  of,  457. 

case  of  equal  roots,  462. 
Haughton,  on  Earth's  density,  108. 


Index. 


525 


Heat,  mechanical  equivalent  of,  477. 

specific,  479. 

latent,  of  liquidity,  502. 

of  vaporization,  503. 

of  expansion,  491. 
Height,  due  to  velocity,  30,  40. 
Helmholtz,  399. 
Herpolhode,  372,  378. 
Herschel,  on  disturbing  forces,  188. 
Him,  505. 
Hodgkinson,   on  laws   of  restitution, 

68. 
Hodograph,  19. 

application  to   focal   orbit,    181, 
182. 
Hooke's  law,  137,  155. 
Huygens,  on  pendulum,  104. 

Ignoration  of  coordinates,  426. 
Impact  and  collision   of  spheres,  66, 
381. 
of  bodies   generally,    280,    288, 
379. 
Impulse,  measure  of,  56. 

in   D'Alembert's  principle,    228, 

287. 
exerted  on  a  fixed  point,  in  rota- 
tion, 367. 
maximum,  281. 
Increase  of  inertia  in  an  oscillating 

system,  469. 
Indicator  diagram,  483. 
Inertia,  law  of,  25,  76. 
forces  of,  59,  227. 
Initial  tensions,  293,  356. 
Instantaneous  centre,  261. 

screw,  334. 
Irreversible  transformations,  495. 
Isentropic  curve,  483. 
Isochronism  of  pendulum,  102. 
Isothermal  curve,  483. 
for  a  perfect  gas,  484. 

Jacobi,   on  motion  in  vertical  circle, 

121. 
Jellett,  50,  386,  394. 
Joule,    on    mechanical   equivalent  of 

heat,  477. 

Kater,  on  determination  of   force  of 

gravity,  144. 
Kepler's  laws,  91,  175. 

modification  of  third  law,  184. 


Kilogrammetre,  126. 

Kinematics,  5,  254. 

Kinetics,  5,  27,  268. 

Kinetic  energy,  133,  416,  419,  453. 

Lagrange,  210,  462. 

on  spherical  pendulum,  216. 

on  small  oscillations,  453. 

generalized  coordinates,  421,  435. 

generalized    equations    for    im- 
pulses, 417. 
Lambert's  theorem,  183. 
Laplace,  referred  to,  332. 
Latent  heat,  of  liquidity,  502. 

of  vaporization,  503. 

of  expansion,  491. 
Laws  of  motion  :  see  Newton. 
Least  action,  436. 
Line  of  quickst  descent,  36. 

M'Cullagh,  on  rotation,  361,  377. 
Mass,  32. 

of  Sun,  186. 
Mean  value  employed,  87. 
Mean  energy  in  vibration,  411. 
Mechanical  equivalent  of  heat,  477. 
Metric  units,  23. 
Minchin,  referred  to,    50,   107,    163, 

265,  272,  341,  343. 
Moment  of  inertia,  137. 
Momental  ellipsoid,  348,  370. 
Momentum,  53. 

estimated  in  any  direction,  74. 

conservation  of,  75,  248. 

moments  of,  243,  246,  273,  286. 

axis,  360,  373. 
Morin's  apparatus,  46,  309. 

on  impulsive  friction,  309. 
Motion,  first  law,  25. 

second  law,  26. 

third  law,  58. 

on  an  inclined  plane,  34,  46,  51. 

parabolic  39. 

of  a  particle,  general  equations  of, 
57. 

of  a  variable  mass,  57. 

in  a  vertical  circle,  99. 

on  a  fixed  curve,  206. 

on  a  fixed  surface,  211. 

of  body  round  fixed  axis,  255. 

round  a  fixed  point,  353,  357. 

of  solid  of  revolution,  389. 
Moving  axes,  22. 


526 


Index. 


Newton,  fluxion  notation,  4,  9,  16. 
referred  to,  76,  153,  176. 
laws  of  motion,  25,  26,  58. 
movable  orbits,  196. 
central  orbits,  166,  171. 
on  coefficient  of  restitution,  68. 
on  resistance  of  medium,  219. 

Orbits,  central,  160. 

movable,  196. 
Oscillation    of    a    simple   pendulum, 
small,  101. 
in  general,  108. 
period  unaffected  by  resistance  of 

air,  123. 
centre  of,  142,  277. 
Oscillations,  small,  445. 

Parabolic  motion,  39,  72,  80. 
Parker,  on  equilibrium  path,  509. 
Peaucellier's  cell,  265. 
Pendulum,  simple,  100. 

compound,  102,  141. 

conical,  115. 

spherical,  212. 

ballistic,  271. 
Percussion,  centre  of,  276. 
Perfect  gas,  481,  484. 
Periodic  time   in   central  orbit,   161, 

175. 
Planetary  perturbations,  185. 
Poinsot,  378. 
Pole  of  rotation,  317. 
Polhode,  372. 
Poncelet,  referred  to,  159. 
Potential,  130,  135. 

energy,  133,  398. 
Poundal  54,  126. 

Principal  axis,  property  of,  in  uniform 
rotation,  97. 

rotation  round,  275. 
Principal  moments,  couple  of,  347. 
Principal  plane,  245. 
Projectile,  parabolic  path  of,  39. 
Pure  rolling,  friction  in,  234. 

Quickest  descent,  line  of,  36. 

Range  of  a  projectile,  41. 
Rankine,  on  steam,  505. 
Eebound  from  a  plane,  69. 
Rectilinear  motion,  25,  147. 
in  resisting  medium,  219. 


Relative  motion,  6,  10. 
Resistance,  of  air,  48. 

does  not  affect  pendulum  period, 
123. 

see  Friction. 
Resisting  medium,  motion  in,  219. 
Restitution,  forces  of,  6Q . 

coefficient  of,  67. 
Reversible    transformations   in   heat, 

483. 
Richer,  observed  retardation  of  pen- 
dulum, 104. 
Rigid  body,  240,  258. 

equations  of  motion  of,  240. 

complete  motion  of,  329. 
Rodrigues,  on  screw  motion,  336. 

coordinates  of,  517. 
Rolling,  pure,  261. 
Rolling  friction  :  see  Couple  of. 
Rotation,  velocity  in,  226. 

acceleration  in,  327. 

of  a  rigid  body,  94. 

of  a  plane  lamina,  95. 

energy  of,  137. 

motion  of,  318. 

instantaneous  centre  of,  261. 
Rotations,  composition  of,  317,  321. 
Routb,    referred  to,    144,    307,    309, 
366,  391,  462,  468. 

on  conjugate  ellipsoid,  363. 

on    equal    factors    of    harmonic- 
determinant,  462. 


Salmon,  referred  to,  378. 

Schell,  referred  to,  279,  343. 

Screw,  axis  and  pitch  of,  333. 
of  resultant  twist,'  337. 

Seconds  pendulum,  102. 
length  of,  123. 

Similar  mechanical  systems,  414. 

Small  oscillation  of  simple  pendulum, 
101,  445. 

Small   oscillations    in    general,    445, 
452. 

Source    and    condensor    in    Camot's 
cycle,  486. 

Space-axes,  330. 

Sphere  used  graphically  in  rotation, 
317. 

Stability  of  motion  of.  small  oscilla- 
tions, 451. 

Statical  measure  of  force,  32. 


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